def get_trace_nullspace_vecs(forward, nullspace, V, V_d, TraceSpace):
"""Gets the nullspace vectors corresponding to the Schur complement
system for the multipliers.
:arg forward: A Slate expression denoting the forward elimination
operator.
:arg nullspace: A nullspace for the original mixed problem
:arg V: The original "unbroken" space.
:arg V_d: The broken space.
:arg TraceSpace: The space of approximate traces.
Returns: A list of vectors describing the nullspace of the multiplier
system
"""
from firedrake import project, assemble, Function
vecs = nullspace.getVecs()
tmp = Function(V)
tmp_b = Function(V_d)
tnsp_tmp = Function(TraceSpace)
forward_action = forward * tmp_b
new_vecs = []
for v in vecs:
with tmp.dat.vec as t:
v.copy(t)
project(tmp, tmp_b)
assemble(forward_action, tensor=tnsp_tmp)
with tnsp_tmp.dat.vec_ro as v:
new_vecs.append(v.copy())
return new_vecs
def test_diagnostic_solver():
Nx, Ny = 32, 32
mesh2d = firedrake.RectangleMesh(Nx, Ny, Lx, Ly)
mesh = firedrake.ExtrudedMesh(mesh2d, layers=1)
Q = firedrake.FunctionSpace(mesh,
family='CG',
degree=2,
vfamily='DG',
vdegree=0)
x, y, ζ = firedrake.SpatialCoordinate(mesh)
h = firedrake.interpolate(h0 - dh * x / Lx, Q)
s = firedrake.interpolate(d + h0 - dh + ds * (1 - x / Lx), Q)
u_expr = firedrake.as_vector(((0.95 + 0.05 * ζ) * exact_u(x), 0))
A = firedrake.Constant(icepack.rate_factor(254.15))
C = firedrake.Constant(0.001)
model = icepack.models.HybridModel()
opts = {'dirichlet_ids': [1, 3, 4], 'tol': 1e-12}
max_degree = 5
Nz = 32
xs = np.array([(Lx / 2, Ly / 2, k / Nz) for k in range(Nz + 1)])
us = np.zeros((max_degree + 1, Nz + 1))
for vdegree in range(max_degree, 0, -1):
solver = icepack.solvers.FlowSolver(model, **opts)
V = firedrake.VectorFunctionSpace(mesh,
dim=2,
family='CG',
degree=2,
vfamily='GL',
vdegree=vdegree)
u0 = firedrake.interpolate(u_expr, V)
u = solver.diagnostic_solve(velocity=u0,
thickness=h,
surface=s,
fluidity=A,
friction=C)
V0 = firedrake.VectorFunctionSpace(mesh,
dim=2,
family='CG',
degree=2,
vfamily='DG',
vdegree=0)
depth_avg_u = firedrake.project(u, V0)
shear_u = firedrake.project(u - depth_avg_u, V)
assert icepack.norm(shear_u, norm_type='Linfty') > 1e-2
us_center = np.array(u.at(xs, tolerance=1e-6))
us[vdegree, :] = np.sqrt(np.sum(us_center**2, 1))
norm = np.linalg.norm(us[max_degree, :])
error = np.linalg.norm(us[vdegree, :] - us[max_degree, :]) / norm
print(error, flush=True)
assert error < 1e-2
def initial_conditions(self):
ic1 = fd.project(fd.Expression([0., 0., 0.]), self.V)
ic2 = fd.project(
fd.Expression(["0.1*(1-cos(pi*x[2]/Lz/2.))", 0., 0.], Lz=self.Lz),
self.V)
# ic3 = fd.project( fd.Expression(["0.1*x[2]/Lz_B", 0., 0.], Lz_B=self.Lz), self.V )
self.X.assign(ic2)
# self.static_solver()
self.U.assign(ic1)
def test_damage_transport():
nx, ny = 32, 32
Lx, Ly = 20e3, 20e3
mesh = firedrake.RectangleMesh(nx, ny, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
V = firedrake.VectorFunctionSpace(mesh, "CG", 2)
Q = firedrake.FunctionSpace(mesh, "CG", 2)
u0 = 100.0
h0, dh = 500.0, 100.0
T = 268.0
ρ = ρ_I * (1 - ρ_I / ρ_W)
Z = icepack.rate_factor(T) * (ρ * g * h0 / 4)**n
q = 1 - (1 - (dh / h0) * (x / Lx))**(n + 1)
du = Z * q * Lx * (h0 / dh) / (n + 1)
u = interpolate(as_vector((u0 + du, 0)), V)
h = interpolate(h0 - dh * x / Lx, Q)
A = firedrake.Constant(icepack.rate_factor(T))
S = firedrake.TensorFunctionSpace(mesh, "DG", 1)
ε = firedrake.project(sym(grad(u)), S)
M = firedrake.project(membrane_stress(strain_rate=ε, fluidity=A), S)
degree = 1
Δ = firedrake.FunctionSpace(mesh, "DG", degree)
D_inflow = firedrake.Constant(0.0)
D = firedrake.Function(Δ)
damage_model = icepack.models.DamageTransport()
damage_solver = icepack.solvers.DamageSolver(damage_model)
final_time = Lx / u0
max_speed = u.at((Lx - 1.0, Ly / 2), tolerance=1e-10)[0]
δx = Lx / nx
timestep = δx / max_speed / (2 * degree + 1)
num_steps = int(final_time / timestep)
dt = final_time / num_steps
for step in range(num_steps):
D = damage_solver.solve(
dt,
damage=D,
velocity=u,
strain_rate=ε,
membrane_stress=M,
damage_inflow=D_inflow,
)
Dmax = D.dat.data_ro[:].max()
assert 0 < Dmax < 1
def initial_conditions(self):
# old: ic1 = fd.project( fd.Expression([0.,0.,0.]), self.V )
# old: ic2 = fd.project( fd.Expression(["0.1*(1-cos(pi*x[2]/Lz/2.))",0.,0.], Lz=self.Lz), self.V )
ic1 = fd.project(fd.Constant([0., 0., 0.]), self.V)
x = fd.SpatialCoordinate(self.mesh)
ic2 = fd.project(
fd.as_vector(
[0.1 * (1 - fd.cos(fd.pi * x[2] / self.Lz / 2.)), 0., 0.]),
self.V)
# ic3 = fd.project( fd.Expression(["0.1*x[2]/Lz_B", 0., 0.], Lz_B=self.Lz), self.V )
self.X.assign(ic2)
# self.static_solver()
self.U.assign(ic1)
def test_eigenvalues():
nx, ny = 32, 32
mesh = firedrake.UnitSquareMesh(nx, ny)
x, y = firedrake.SpatialCoordinate(mesh)
V = firedrake.VectorFunctionSpace(mesh, family='CG', degree=2)
u = interpolate(as_vector((x, 0)), V)
Q = firedrake.FunctionSpace(mesh, family='DG', degree=2)
ε = sym(grad(u))
Λ1, Λ2 = eigenvalues(ε)
λ1 = firedrake.project(Λ1, Q)
λ2 = firedrake.project(Λ2, Q)
assert norm(λ1 - Constant(1)) < norm(u) / (nx * ny)
assert norm(λ2) < norm(u) / (nx * ny)
def test_converting_fields():
δT = 5.0
T_surface = Tm - δT
T_expr = firedrake.min_value(Tm, T_surface + 2 * δT * (1 - ζ))
f_expr = firedrake.max_value(0, 0.0033 * (1 - 2 * ζ))
model = icepack.models.HeatTransport3D()
E = firedrake.project(model.energy_density(T_expr, f_expr), Q)
f = firedrake.project(model.meltwater_fraction(E), Q)
T = firedrake.project(model.temperature(E), Q)
avg_meltwater = firedrake.assemble(f * ds_b) / (Lx * Ly)
assert avg_meltwater > 0
avg_temp = firedrake.assemble(T * h * dx) / firedrake.assemble(h * dx)
assert (avg_temp > T_surface) and (avg_temp < Tm)
def __init__(self, V, gcur, u0, u0_mult, i, t, dt):
if V.component is not None: # bottommost space is bit of VFS
if V.parent.index is None: # but not part of a MFS
sub = V.component
try:
gdat = interpolate(gcur - u0_mult[i] * u0, V)
gmethod = lambda g, u: gdat.interpolate(g - u0_mult[i] * u.
sub(sub))
except: # noqa: E722
gdat = project(gcur - u0_mult[i] * u0, V)
gmethod = lambda g, u: gdat.project(g - u0_mult[i] * u.sub(
sub))
else: # V is a bit of a VFS inside an MFS
sub0 = V.parent.index
sub1 = V.component
try:
gdat = interpolate(
gcur - u0_mult[i] * u0.sub(sub0).sub(sub1), V)
gmethod = lambda g, u: gdat.interpolate(g - u0_mult[
i] * u.sub(sub0).sub(sub1))
except: # noqa: E722
gdat = project(gcur - u0_mult[i] * u0.sub(sub0).sub(sub1),
V)
gmethod = lambda g, u: gdat.project(g - u0_mult[i] * u.sub(
sub0).sub(sub1))
else: # V is not a bit of a VFS
if V.index is None: # not part of MFS, either
try:
gdat = interpolate(gcur - u0_mult[i] * u0, V)
gmethod = lambda g, u: gdat.interpolate(g - u0_mult[i] * u)
except: # noqa: E722
gdat = project(gcur - u0_mult[i] * u0, V)
gmethod = lambda g, u: gdat.project(g - u0_mult[i] * u)
else: # part of MFS
sub = V.index
try:
gdat = interpolate(gcur - u0_mult[i] * u0.sub(sub), V)
gmethod = lambda g, u: gdat.interpolate(g - u0_mult[i] * u.
sub(sub))
except: # noqa: E722
gdat = project(gcur - u0_mult[i] * u0.sub(sub), V)
gmethod = lambda g, u: gdat.project(g - u0_mult[i] * u.sub(
sub))
self.gstuff = (gdat, gcur, gmethod)
def apply(self, pc, x, y):
"""We solve the forward eliminated problem for the
approximate traces of the scalar solution (the multipliers)
and reconstruct the "broken flux and scalar variable."
Lastly, we project the broken solutions into the mimetic
non-broken finite element space.
"""
from firedrake import project
# Transfer non-broken x into a firedrake function
with self.unbroken_rhs.dat.vec as v:
x.copy(v)
# Transfer unbroken_rhs into broken_rhs
field0, field1 = self.unbroken_rhs.split()
bfield0, bfield1 = self.broken_rhs.split()
# This updates broken_rhs
project(field0, bfield0)
field1.dat.copy(bfield1.dat)
# Compute the rhs for the multiplier system
self._assemble_Srhs()
# Solve the system for the Lagrange multipliers
with self.schur_rhs.dat.vec_ro as b:
with self.trace_solution.dat.vec as x:
self.ksp.solve(b, x)
# Assemble the pressure and velocity (in that order)
# using the Lagrange multipliers
self._assemble_pressure()
self._assemble_velocity()
# Project the broken solution into non-broken spaces
sigma_h, u_h = self.broken_solution.split()
sigma_u, u_u = self.unbroken_solution.split()
u_h.dat.copy(u_u.dat)
self.projector.project()
with self.unbroken_solution.dat.vec_ro as v:
v.copy(y)
def mult(self, mat, x, y):
# Copy function data into the fivredrake function
self.x_fntn.dat.data[:] = x[:]
# Transfer the function to meshmode
self.meshmode_src_connection.from_firedrake(project(
self.x_fntn, dgfspace),
out=self.x_mm_fntn)
# Restrict to boundary
x_mm_fntn_on_bdy = src_bdy_connection(self.x_mm_fntn)
# Apply the operation
potential_int_mm = self.pyt_op(self.actx,
u=x_mm_fntn_on_bdy,
k=self.k)
grad_potential_int_mm = self.pyt_grad_op(self.actx,
u=x_mm_fntn_on_bdy,
k=self.k)
# Store in firedrake
self.potential_int.dat.data[target_indices] = potential_int_mm.get(
)
for dim in range(grad_potential_int_mm.shape[0]):
self.grad_potential_int.dat.data[
target_indices, dim] = grad_potential_int_mm[dim].get()
# Integrate the potential
r"""
Compute the inner products using firedrake. Note this
will be subtracted later, hence appears off by a sign.
.. math::
\langle
n(x) \cdot \nabla(
\int_\Gamma(
u(y) \partial_n H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
), v
\rangle_\Sigma
- \langle
i \kappa \cdot
\int_\Gamma(
u(y) \partial_n H_0^{(1)}(\kappa |x - y|)
)d\gamma(y), v
\rangle_\Sigma
"""
self.pyt_result = assemble(
inner(inner(self.grad_potential_int, self.n), self.v) *
ds(outer_bdy_id) -
inner(self.potential_int, self.v) * ds(outer_bdy_id))
# y <- Ax - evaluated potential
self.A.mult(x, y)
with self.pyt_result.dat.vec_ro as ep:
y.axpy(-1, ep)
def siahorizontalvelocity(mesh):
hbase = surfaceelevation(mesh)
if mesh._base_mesh.cell_dimension() == 2:
if mesh._base_mesh.ufl_cell() == fd.quadrilateral:
Vvectorbase = fd.VectorFunctionSpace(mesh._base_mesh, 'DQ', 0)
VvectorR = fd.VectorFunctionSpace(mesh,
'DQ',
0,
vfamily='R',
vdegree=0,
dim=2)
else:
Vvectorbase = fd.VectorFunctionSpace(mesh._base_mesh, 'DP', 0)
VvectorR = fd.VectorFunctionSpace(mesh,
'DP',
0,
vfamily='R',
vdegree=0,
dim=2)
gradhbase = fd.project(fd.grad(hbase), Vvectorbase)
Vvector = fd.VectorFunctionSpace(mesh, 'DQ', 0, dim=2)
elif mesh._base_mesh.cell_dimension() == 1:
Vvectorbase = fd.FunctionSpace(mesh._base_mesh, 'DP', 0)
gradhbase = fd.project(hbase.dx(0), Vvectorbase)
VvectorR = fd.FunctionSpace(mesh, 'DP', 0, vfamily='R', vdegree=0)
Vvector = fd.FunctionSpace(mesh, 'DQ', 0)
else:
raise ValueError('base mesh of extruded input mesh must be 1D or 2D')
gradh = fd.Function(VvectorR)
gradh.dat.data[:] = gradhbase.dat.data_ro[:]
h = extend(mesh, hbase)
DQ0 = fd.FunctionSpace(mesh, 'DQ', 0)
h0 = fd.project(h, DQ0)
x = fd.SpatialCoordinate(mesh)
z0 = fd.project(x[mesh._base_mesh.cell_dimension()], DQ0)
# FIXME following only valid in flat bed case
uvsia = -Gamma * (h0**(n + 1) -
(h0 - z0)**(n + 1)) * abs(gradh)**(n - 1) * gradh
uv = fd.Function(Vvector).interpolate(uvsia)
uv.rename('velocitySIA')
return uv
def create_schur_nullspace(P, forward, V, V_d, TraceSpace, comm):
"""Gets the nullspace vectors corresponding to the Schur complement
system for the multipliers.
:arg P: The mixed operator from the ImplicitMatrixContext.
:arg forward: A Slate expression denoting the forward elimination
operator.
:arg V: The original "unbroken" space.
:arg V_d: The broken space.
:arg TraceSpace: The space of approximate traces.
Returns: A nullspace (if there is one) for the Schur-complement system.
"""
from firedrake import assemble, Function, project
nullspace = P.getNullSpace()
if nullspace.handle == 0:
# No nullspace
return None
vecs = nullspace.getVecs()
tmp = Function(V)
tmp_b = Function(V_d)
tnsp_tmp = Function(TraceSpace)
forward_action = forward * tmp_b
new_vecs = []
for v in vecs:
with tmp.dat.vec_wo as t:
v.copy(t)
project(tmp, tmp_b)
assemble(forward_action, tensor=tnsp_tmp)
with tnsp_tmp.dat.vec_ro as v:
new_vecs.append(v.copy())
# Normalize
for v in new_vecs:
v.normalize()
schur_nullspace = PETSc.NullSpace().create(vectors=new_vecs, comm=comm)
return schur_nullspace
def tripcolor(function, *args, **kwargs):
r"""Plot a discontinuous finite element field"""
axes = kwargs.pop('axes', plt.gca())
mesh = function.ufl_domain()
coordinates = _get_coordinates(mesh)
coords = coordinates.dat.data_ro
x, y = coords[:, 0], coords[:, 1]
triangles = coordinates.cell_node_map().values
triangulation = matplotlib.tri.Triangulation(x, y, triangles)
Q = firedrake.FunctionSpace(mesh, family='DG', degree=0)
fn = firedrake.project(function, Q)
return axes.tripcolor(triangulation, fn.dat.data_ro, *args, **kwargs)
def wave(n, deg, butcher_tableau, splitting=AI):
N = 2**n
msh = UnitIntervalMesh(N)
params = {
"snes_type": "ksponly",
"ksp_type": "preonly",
"mat_type": "aij",
"pc_type": "lu"
}
V = FunctionSpace(msh, "CG", deg)
W = FunctionSpace(msh, "DG", deg - 1)
Z = V * W
x, = SpatialCoordinate(msh)
t = Constant(0.0)
dt = Constant(2.0 / N)
up = project(as_vector([0, sin(pi * x)]), Z)
u, p = split(up)
v, w = TestFunctions(Z)
F = (inner(Dt(u), v) * dx + inner(u.dx(0), w) * dx + inner(Dt(p), w) * dx -
inner(p, v.dx(0)) * dx)
E = 0.5 * (inner(u, u) * dx + inner(p, p) * dx)
stepper = TimeStepper(F,
butcher_tableau,
t,
dt,
up,
solver_parameters=params,
splitting=splitting)
energies = []
while (float(t) < 1.0):
if (float(t) + float(dt) > 1.0):
dt.assign(1.0 - float(t))
stepper.advance()
t.assign(float(t) + float(dt))
energies.append(assemble(E))
return np.array(energies)
def heat(n, deg, time_stages, stage_type="deriv", splitting=IA):
N = 2**n
msh = UnitIntervalMesh(N)
params = {
"snes_type": "ksponly",
"ksp_type": "preonly",
"mat_type": "aij",
"pc_type": "lu"
}
V = FunctionSpace(msh, "CG", deg)
x, = SpatialCoordinate(msh)
t = Constant(0.0)
dt = Constant(2.0 / N)
uexact = exp(-t) * sin(pi * x)
rhs = expand_derivatives(diff(uexact, t)) - div(grad(uexact))
butcher_tableau = GaussLegendre(time_stages)
u = project(uexact, V)
v = TestFunction(V)
F = (inner(Dt(u), v) * dx + inner(grad(u), grad(v)) * dx -
inner(rhs, v) * dx)
bc = DirichletBC(V, Constant(0), "on_boundary")
stepper = TimeStepper(F,
butcher_tableau,
t,
dt,
u,
bcs=bc,
solver_parameters=params,
stage_type=stage_type,
splitting=splitting)
while (float(t) < 1.0):
if (float(t) + float(dt) > 1.0):
dt.assign(1.0 - float(t))
stepper.advance()
t.assign(float(t) + float(dt))
return errornorm(uexact, u) / norm(uexact)
def to_2nd_order(mesh, circle_bdy_id=None, rad=1.0):
# make new coordinates function
V = VectorFunctionSpace(mesh, 'CG', 2)
new_coordinates = project(mesh.coordinates, V)
# If we have a circle, move any nodes on the circle bdy
# onto the circle. Note circle MUST be centered at origin
if circle_bdy_id is not None:
nodes_on_circle = V.boundary_nodes(circle_bdy_id, 'geometric')
#Force all cell nodes to have given radius :arg:`rad`
for node in nodes_on_circle:
scale = rad / la.norm(new_coordinates.dat.data[node])
new_coordinates.dat.data[node] *= scale
# Make a new mesh with given coordinates
return Mesh(new_coordinates)
def test_strain_heating():
E_initial = firedrake.interpolate(E_surface + q_bed / α * h * (1 - ζ), Q)
u0 = 100.0
du = 100.0
u_expr = as_vector((u0 + du * x / Lx, 0))
u = firedrake.interpolate(u_expr, V)
w = firedrake.interpolate((-du / Lx + dh / Lx / h * u[0]) * ζ, W)
E_q = E_initial.copy(deepcopy=True)
E_0 = E_initial.copy(deepcopy=True)
from icepack.models.hybrid import (
horizontal_strain,
vertical_strain,
stresses
)
model = icepack.models.HeatTransport3D()
T = model.temperature(E_q)
A = icepack.rate_factor(T)
ε_x, ε_z = horizontal_strain(u, s, h), vertical_strain(u, h)
τ_x, τ_z = stresses(ε_x, ε_z, A)
q = firedrake.project(inner(τ_x, ε_x) + inner(τ_z, ε_z), Q)
fields = {
'velocity': u,
'vertical_velocity': w,
'thickness': h,
'surface': s,
'heat_bed': Constant(q_bed),
'energy_inflow': E_initial,
'energy_surface': Constant(E_surface)
}
dt = 10.0
final_time = Lx / u0
num_steps = int(final_time / dt) + 1
solver_strain = icepack.solvers.HeatTransportSolver(model)
solver_no_strain = icepack.solvers.HeatTransportSolver(model)
for step in range(num_steps):
E_q = solver_strain.solve(dt, energy=E_q, heat=q, **fields)
E_0 = solver_no_strain.solve(dt, energy=E_0, heat=Constant(0), **fields)
assert assemble(E_q * h * dx) > assemble(E_0 * h * dx)
def test_strain_heating():
E_initial = firedrake.interpolate(E_surface + q_bed / α * h * (1 - ζ), Q)
u0 = 100.0
du = 100.0
u_expr = as_vector((u0 + du * x / Lx, 0))
u = firedrake.interpolate(u_expr, V)
w = firedrake.interpolate((-du / Lx + dh / Lx / h * u[0]) * ζ, W)
E_q = E_initial.copy(deepcopy=True)
E_0 = E_initial.copy(deepcopy=True)
model = icepack.models.HeatTransport3D()
T = model.temperature(E_q)
A = icepack.rate_factor(T)
ε_x = horizontal_strain_rate(velocity=u, surface=s, thickness=h)
ε_z = vertical_strain_rate(velocity=u, thickness=h)
τ_x, τ_z = stresses(strain_rate_x=ε_x, strain_rate_z=ε_z, fluidity=A)
q = firedrake.project(inner(τ_x, ε_x) + inner(τ_z, ε_z), Q)
fields = {
"velocity": u,
"vertical_velocity": w,
"thickness": h,
"surface": s,
"heat_bed": Constant(q_bed),
"energy_inflow": E_initial,
"energy_surface": Constant(E_surface),
}
dt = 10.0
final_time = Lx / u0
num_steps = int(final_time / dt) + 1
solver_strain = icepack.solvers.HeatTransportSolver(model)
solver_no_strain = icepack.solvers.HeatTransportSolver(model)
for step in range(num_steps):
E_q = solver_strain.solve(dt, energy=E_q, heat=q, **fields)
E_0 = solver_no_strain.solve(dt,
energy=E_0,
heat=Constant(0),
**fields)
assert assemble(E_q * h * dx) > assemble(E_0 * h * dx)
def test_inflow_boundary(scheme):
start = 16
finish = 3 * start
incr = 4
num_points = np.array(list(range(start, finish + incr, incr)))
errors = np.zeros_like(num_points, dtype=np.float64)
for k, nx in enumerate(num_points):
mesh = firedrake.UnitSquareMesh(nx, nx, diagonal='crossed')
degree = 1
Q = firedrake.FunctionSpace(mesh, 'DG', 1)
x = firedrake.SpatialCoordinate(mesh)
U = Constant(1.)
u = as_vector((U, 0.))
q_in = Constant(1.)
s = Constant(0.)
final_time = 0.5
min_diameter = mesh.cell_sizes.dat.data_ro[:].min()
max_speed = 1.
multiplier = multipliers[scheme] / 2
timestep = multiplier * min_diameter / max_speed / (2 * degree + 1)
num_steps = int(final_time / timestep)
dt = final_time / num_steps
q_0 = firedrake.project(q_in - x[0], Q)
equation = plumes.models.advection.make_equation(u, s, q_in)
integrator = scheme(equation, q_0)
for step in range(num_steps):
integrator.step(dt)
q = integrator.state
T = Constant(final_time)
expr = q_in - firedrake.conditional(x[0] > U * T, x[0] - U * T, 0)
errors[k] = assemble(abs(q - expr) * dx) / assemble(abs(expr) * dx)
slope, intercept = np.polyfit(np.log2(1 / num_points), np.log2(errors), 1)
print(f'log(error) ~= {slope:5.3f} * log(dx) {intercept:+5.3f}')
assert slope > degree - 0.1
def betaInit(s, h, speed, V, Q, Q1, grounded, inversionParams):
"""Compute intitial beta using 0.5 taud.
Parameters
----------
s : firedrake function
model surface elevation
h : firedrake function
model thickness
speed : firedrake function
modelled speed
V : firedrake vector function space
vector function space
Q : firedrake function space
scalar function space
grounded : firedrake function
Mask with 1s for grounded 0 for floating.
"""
# Use a result from prior inversion
checkFile = inversionParams['initFile']
Quse = Q
if inversionParams['initWithDeg1']:
checkFile = f'{inversionParams["inversionResult"]}.deg1'
Quse = Q1
if checkFile is not None:
betaTemp = mf.getCheckPointVars(checkFile, 'betaInv', Quse)['betaInv']
beta1 = icepack.interpolate(betaTemp, Q)
return beta1
# No prior result, so use fraction of taud
tauD = firedrake.project(-rhoI * g * h * grad(s), V)
#
stress = firedrake.sqrt(firedrake.inner(tauD, tauD))
Print('stress', firedrake.assemble(stress * firedrake.dx))
fraction = firedrake.Constant(0.95)
U = max_value(speed, 1)
C = fraction * stress / U**(1/m)
if inversionParams['friction'] == 'schoof':
mExp = 1/m + 1
U0 = firedrake.Constant(inversionParams['uThresh'])
C = C * (m/(m+1)) * (U0**mExp + U**mExp)**(1/(m+1))
beta = firedrake.interpolate(firedrake.sqrt(C) * grounded, Q)
return beta
def regularization_form(r):
mesh = UnitSquareMesh(2 ** r, 2 ** r)
x = SpatialCoordinate(mesh)
S = VectorFunctionSpace(mesh, "CG", 1)
beta = 4.0
reg_solver = RegularizationSolver(S, mesh, beta=beta, gamma=0.0, dx=dx)
# Exact solution with free Neumann boundary conditions for this domain
u_exact = Function(S)
u_exact_component = cos(x[0] * pi * 2) * cos(x[1] * pi * 2)
u_exact.interpolate(as_vector((u_exact_component, u_exact_component)))
f = Function(S)
theta = TestFunction(S)
f_component = (1 + beta * 8 * pi * pi) * u_exact_component
f.interpolate(as_vector((f_component, f_component)))
rhs_form = inner(f, theta) * dx
velocity = Function(S)
rhs = assemble(rhs_form)
reg_solver.solve(velocity, rhs)
File("solution_vel_unitsquare.pvd").write(velocity)
return norm(project(u_exact - velocity, S))
def test_2D_dirichlet_regions():
# Can't do mesh refinement because MeshHierarchy ruins the domain tags
mesh = Mesh("./2D_mesh.msh")
dim = mesh.geometric_dimension()
x = SpatialCoordinate(mesh)
S = VectorFunctionSpace(mesh, "CG", 1)
beta = 1.0
reg_solver = RegularizationSolver(
S, mesh, beta=beta, gamma=0.0, dx=dx, design_domain=0
)
# Exact solution with free Neumann boundary conditions for this domain
u_exact = Function(S)
u_exact_component = (-cos(x[0] * pi * 2) + 1) * (-cos(x[1] * pi * 2) + 1)
u_exact.interpolate(
as_vector(tuple(u_exact_component for _ in range(dim)))
)
f = Function(S)
f_component = (
-beta
* (
4.0 * pi * pi * cos(2 * pi * x[0]) * (-cos(2 * pi * x[1]) + 1)
+ 4.0 * pi * pi * cos(2 * pi * x[1]) * (-cos(2 * pi * x[0]) + 1)
)
+ u_exact_component
)
f.interpolate(as_vector(tuple(f_component for _ in range(dim))))
theta = TestFunction(S)
rhs_form = inner(f, theta) * dx
velocity = Function(S)
rhs = assemble(rhs_form)
reg_solver.solve(velocity, rhs)
assert norm(project(domainify(u_exact, x) - velocity, S)) < 1e-3
def depth_average(q3d, weight=firedrake.Constant(1)):
r"""Return the weighted depth average of a function on an extruded mesh"""
element3d = q3d.ufl_element()
# Create the element `E x DG0` where `E` is the horizontal element for the
# input field
element_z = firedrake.FiniteElement(family='DG', cell='interval', degree=0)
shape = q3d.ufl_shape
if len(shape) == 0:
element_xy = element3d.sub_elements()[0]
element_avg = firedrake.TensorProductElement(element_xy, element_z)
element2d = element_xy
elif len(shape) == 1:
element_xy = element3d.sub_elements()[0].sub_elements()[0]
element_u = firedrake.TensorProductElement(element_xy, element_z)
element_avg = firedrake.VectorElement(element_u, dim=shape[0])
element2d = firedrake.VectorElement(element_xy, dim=shape[0])
else:
raise NotImplementedError('Depth average of tensor fields not yet '
'implemented!')
# Project the weighted 3D field onto vertical DG0
mesh3d = q3d.ufl_domain()
Q_avg = firedrake.FunctionSpace(mesh3d, element_avg)
q_avg = firedrake.project(weight * q3d, Q_avg)
# Create a function space on the 2D mesh and a 2D function defined on this
# space, then copy the raw vector of expansion coefficients from the 3D DG0
# field into the coefficients of the 2D field. TODO: Get some assurance
# from the firedrake folks that this will always work.
mesh2d = mesh3d._base_mesh
Q2D = firedrake.FunctionSpace(mesh2d, element2d)
q2d = firedrake.Function(Q2D)
q2d.dat.data[:] = q_avg.dat.data_ro[:]
return q2d
def nonlocal_integral_eq(
mesh,
scatterer_bdy_id,
outer_bdy_id,
wave_number,
options_prefix=None,
solver_parameters=None,
fspace=None,
vfspace=None,
true_sol_grad_expr=None,
actx=None,
dgfspace=None,
dgvfspace=None,
meshmode_src_connection=None,
qbx_kwargs=None,
):
r"""
see run_method for descriptions of unlisted args
args:
gamma and beta are used to precondition
with the following equation:
\Delta u - \kappa^2 \gamma u = 0
(\partial_n - i\kappa\beta) u |_\Sigma = 0
"""
# make sure we get outer bdy id as tuple in case it consists of multiple ids
if isinstance(outer_bdy_id, int):
outer_bdy_id = [outer_bdy_id]
outer_bdy_id = tuple(outer_bdy_id)
# away from the excluded region, but firedrake and meshmode point
# into
pyt_inner_normal_sign = -1
ambient_dim = mesh.geometric_dimension()
# {{{ Build src and tgt
# build connection meshmode near src boundary -> src boundary inside meshmode
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
from meshmode.discretization.connection import make_face_restriction
factory = InterpolatoryQuadratureSimplexGroupFactory(
dgfspace.finat_element.degree)
src_bdy_connection = make_face_restriction(actx,
meshmode_src_connection.discr,
factory, scatterer_bdy_id)
# source is a qbx layer potential
from pytential.qbx import QBXLayerPotentialSource
disable_refinement = (fspace.mesh().geometric_dimension() == 3)
qbx = QBXLayerPotentialSource(src_bdy_connection.to_discr,
**qbx_kwargs,
_disable_refinement=disable_refinement)
# get target indices and point-set
target_indices, target = get_target_points_and_indices(
fspace, outer_bdy_id)
# }}}
# build the operations
from pytential import bind, sym
r"""
..math:
x \in \Sigma
grad_op(x) =
\nabla(
\int_\Gamma(
u(y) \partial_n H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
)
"""
grad_op = pyt_inner_normal_sign * sym.grad(
ambient_dim,
sym.D(HelmholtzKernel(ambient_dim),
sym.var("u"),
k=sym.var("k"),
qbx_forced_limit=None))
r"""
..math:
x \in \Sigma
op(x) =
i \kappa \cdot
\int_\Gamma(
u(y) \partial_n H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
"""
op = pyt_inner_normal_sign * 1j * sym.var("k") * (sym.D(
HelmholtzKernel(ambient_dim),
sym.var("u"),
k=sym.var("k"),
qbx_forced_limit=None))
# bind the operations
pyt_grad_op = bind((qbx, target), grad_op)
pyt_op = bind((qbx, target), op)
# }}}
class MatrixFreeB(object):
def __init__(self, A, pyt_grad_op, pyt_op, actx, kappa):
"""
:arg kappa: The wave number
"""
self.actx = actx
self.k = kappa
self.pyt_op = pyt_op
self.pyt_grad_op = pyt_grad_op
self.A = A
self.meshmode_src_connection = meshmode_src_connection
# {{{ Create some functions needed for multing
self.x_fntn = Function(fspace)
# CG
self.potential_int = Function(fspace)
self.potential_int.dat.data[:] = 0.0
self.grad_potential_int = Function(vfspace)
self.grad_potential_int.dat.data[:] = 0.0
self.pyt_result = Function(fspace)
self.n = FacetNormal(mesh)
self.v = TestFunction(fspace)
# some meshmode ones
self.x_mm_fntn = self.meshmode_src_connection.discr.empty(
self.actx, dtype='c')
# }}}
def mult(self, mat, x, y):
# Copy function data into the fivredrake function
self.x_fntn.dat.data[:] = x[:]
# Transfer the function to meshmode
self.meshmode_src_connection.from_firedrake(project(
self.x_fntn, dgfspace),
out=self.x_mm_fntn)
# Restrict to boundary
x_mm_fntn_on_bdy = src_bdy_connection(self.x_mm_fntn)
# Apply the operation
potential_int_mm = self.pyt_op(self.actx,
u=x_mm_fntn_on_bdy,
k=self.k)
grad_potential_int_mm = self.pyt_grad_op(self.actx,
u=x_mm_fntn_on_bdy,
k=self.k)
# Store in firedrake
self.potential_int.dat.data[target_indices] = potential_int_mm.get(
)
for dim in range(grad_potential_int_mm.shape[0]):
self.grad_potential_int.dat.data[
target_indices, dim] = grad_potential_int_mm[dim].get()
# Integrate the potential
r"""
Compute the inner products using firedrake. Note this
will be subtracted later, hence appears off by a sign.
.. math::
\langle
n(x) \cdot \nabla(
\int_\Gamma(
u(y) \partial_n H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
), v
\rangle_\Sigma
- \langle
i \kappa \cdot
\int_\Gamma(
u(y) \partial_n H_0^{(1)}(\kappa |x - y|)
)d\gamma(y), v
\rangle_\Sigma
"""
self.pyt_result = assemble(
inner(inner(self.grad_potential_int, self.n), self.v) *
ds(outer_bdy_id) -
inner(self.potential_int, self.v) * ds(outer_bdy_id))
# y <- Ax - evaluated potential
self.A.mult(x, y)
with self.pyt_result.dat.vec_ro as ep:
y.axpy(-1, ep)
# {{{ Compute normal helmholtz operator
u = TrialFunction(fspace)
v = TestFunction(fspace)
r"""
.. math::
\langle
\nabla u, \nabla v
\rangle
- \kappa^2 \cdot \langle
u, v
\rangle
- i \kappa \langle
u, v
\rangle_\Sigma
"""
a = inner(grad(u), grad(v)) * dx \
- Constant(wave_number**2) * inner(u, v) * dx \
- Constant(1j * wave_number) * inner(u, v) * ds(outer_bdy_id)
# get the concrete matrix from a general bilinear form
A = assemble(a).M.handle
# }}}
# {{{ Setup Python matrix
B = PETSc.Mat().create()
# build matrix context
Bctx = MatrixFreeB(A, pyt_grad_op, pyt_op, actx, wave_number)
# set up B as same size as A
B.setSizes(*A.getSizes())
B.setType(B.Type.PYTHON)
B.setPythonContext(Bctx)
B.setUp()
# }}}
# {{{ Create rhs
# Remember f is \partial_n(true_sol)|_\Gamma
# so we just need to compute \int_\Gamma\partial_n(true_sol) H(x-y)
sigma = sym.make_sym_vector("sigma", ambient_dim)
r"""
..math:
x \in \Sigma
grad_op(x) =
\nabla(
\int_\Gamma(
f(y) H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
)
"""
grad_op = pyt_inner_normal_sign * \
sym.grad(ambient_dim, sym.S(HelmholtzKernel(ambient_dim),
sym.n_dot(sigma),
k=sym.var("k"), qbx_forced_limit=None))
r"""
..math:
x \in \Sigma
op(x) =
i \kappa \cdot
\int_\Gamma(
f(y) H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
)
"""
op = 1j * sym.var("k") * pyt_inner_normal_sign * \
sym.S(HelmholtzKernel(ambient_dim),
sym.n_dot(sigma),
k=sym.var("k"),
qbx_forced_limit=None)
rhs_grad_op = bind((qbx, target), grad_op)
rhs_op = bind((qbx, target), op)
# Transfer to meshmode
metadata = {'quadrature_degree': 2 * fspace.ufl_element().degree()}
dg_true_sol_grad = project(true_sol_grad_expr,
dgvfspace,
form_compiler_parameters=metadata)
true_sol_grad_mm = meshmode_src_connection.from_firedrake(dg_true_sol_grad,
actx=actx)
true_sol_grad_mm = src_bdy_connection(true_sol_grad_mm)
# Apply the operations
f_grad_convoluted_mm = rhs_grad_op(actx,
sigma=true_sol_grad_mm,
k=wave_number)
f_convoluted_mm = rhs_op(actx, sigma=true_sol_grad_mm, k=wave_number)
# Transfer function back to firedrake
f_grad_convoluted = Function(vfspace)
f_convoluted = Function(fspace)
f_grad_convoluted.dat.data[:] = 0.0
f_convoluted.dat.data[:] = 0.0
for dim in range(f_grad_convoluted_mm.shape[0]):
f_grad_convoluted.dat.data[target_indices,
dim] = f_grad_convoluted_mm[dim].get()
f_convoluted.dat.data[target_indices] = f_convoluted_mm.get()
r"""
\langle
f, v
\rangle_\Gamma
+ \langle
i \kappa \cdot \int_\Gamma(
f(y) H_0^{(1)}(\kappa |x - y|)
)d\gamma(y), v
\rangle_\Sigma
- \langle
n(x) \cdot \nabla(
\int_\Gamma(
f(y) H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
), v
\rangle_\Sigma
"""
rhs_form = inner(inner(true_sol_grad_expr, FacetNormal(mesh)),
v) * ds(scatterer_bdy_id, metadata=metadata) \
+ inner(f_convoluted, v) * ds(outer_bdy_id) \
- inner(inner(f_grad_convoluted, FacetNormal(mesh)),
v) * ds(outer_bdy_id)
rhs = assemble(rhs_form)
# {{{ set up a solver:
solution = Function(fspace, name="Computed Solution")
# {{{ Used for preconditioning
if 'gamma' in solver_parameters or 'beta' in solver_parameters:
gamma = complex(solver_parameters.pop('gamma', 1.0))
import cmath
beta = complex(solver_parameters.pop('beta', cmath.sqrt(gamma)))
p = inner(grad(u), grad(v)) * dx \
- Constant(wave_number**2 * gamma) * inner(u, v) * dx \
- Constant(1j * wave_number * beta) * inner(u, v) * ds(outer_bdy_id)
P = assemble(p).M.handle
else:
P = A
# }}}
# Set up options to contain solver parameters:
ksp = PETSc.KSP().create()
if solver_parameters['pc_type'] == 'pyamg':
del solver_parameters['pc_type'] # We are using the AMG preconditioner
pyamg_tol = solver_parameters.get('pyamg_tol', None)
if pyamg_tol is not None:
pyamg_tol = float(pyamg_tol)
pyamg_maxiter = solver_parameters.get('pyamg_maxiter', None)
if pyamg_maxiter is not None:
pyamg_maxiter = int(pyamg_maxiter)
ksp.setOperators(B)
ksp.setUp()
pc = ksp.pc
pc.setType(pc.Type.PYTHON)
pc.setPythonContext(
AMGTransmissionPreconditioner(wave_number,
fspace,
A,
tol=pyamg_tol,
maxiter=pyamg_maxiter,
use_plane_waves=True))
# Otherwise use regular preconditioner
else:
ksp.setOperators(B, P)
options_manager = OptionsManager(solver_parameters, options_prefix)
options_manager.set_from_options(ksp)
import petsc4py.PETSc
petsc4py.PETSc.Sys.popErrorHandler()
with rhs.dat.vec_ro as b:
with solution.dat.vec as x:
ksp.solve(b, x)
# }}}
return ksp, solution
#
# Now we're ready to solve the variational problem. We define `w` to be a
# function to hold the solution on the mixed space.
w = fd.Function(W)
# solve
fd.solve(a == L, w, bcs=[bc0, bc1])
# %%
# Post-process and plot interesting fields
print('* post-process')
# collect fields
vel, press = w.split()
press.rename('pressure')
# Projecting velocity field to a continuous space (visualization purporse)
P1 = fd.VectorFunctionSpace(mesh, "CG", 1)
vel_proj = fd.project(vel, P1)
vel_proj.rename('velocity_proj')
# project permeability Kxx
kxx = fd.Function(DG)
kxx.rename('Kxx')
kxx.dat.data[...] = 1 / Kinv.dat.data[:, 0, 0]
# print results
fd.File("plots/darcy_rt.pvd").write(vel_proj, press, kxx)
print('* normal termination')
def test_linear_bed(scheme):
start = 16
finish = 2 * start
incr = 4
num_points = np.array(list(range(start, finish + incr, incr)))
errors = np.zeros_like(num_points, dtype=np.float64)
# TODO: Test with different degrees + function spaces
degree = 1
for k, nx in enumerate(num_points):
Lx, Ly = 20.0, 20.0
mesh = firedrake.RectangleMesh(nx, nx, Lx, Ly, diagonal='crossed')
Q = firedrake.FunctionSpace(mesh, 'DG', degree)
V = firedrake.VectorFunctionSpace(mesh, 'DG', degree)
Z = Q * V
x = firedrake.SpatialCoordinate(mesh)
b_0 = Constant(0.2)
δb = Constant(0.0)
b = b_0 - δb * x[0] / Lx
g = Constant(gravity)
h_in = Constant(0.1)
u_in = Constant(2.)
q_in = h_in * u_in
# Make the exact steady-state thickness.
h = firedrake.project(h_in, Q)
φ = firedrake.TestFunction(Q)
F = (h + 0.5 * q_in**2 / (g * h**2) -
(h_in + 0.5 * q_in**2 / (g * h_in**2)) - δb * x[0] / Lx) * φ * dx
firedrake.solve(F == 0, h)
# The exact steady-state momentum is a constant.
q = firedrake.project(as_vector((q_in, 0.0)), V)
# Get the maximum wave speed and choose a timestep that will satisfy
# the CFL condition.
c = firedrake.project(q[0] / h + sqrt(g * h), Q)
max_speed = c.dat.data_ro[:].max()
min_diameter = mesh.cell_sizes.dat.data_ro[:].min()
multiplier = multipliers[scheme]
timestep = multiplier * min_diameter / max_speed / (2 * degree + 1)
# Run the simulation for a full residence time of the system.
final_time = 2 * Lx / float(u_in)
num_steps = int(final_time / timestep)
dt = final_time / num_steps
z_0 = firedrake.Function(Z)
h_0, q_0 = z_0.split()
q_0.assign(q)
# Add a small perturbation to the initial state.
δh = Constant(h_in / 10)
r = Constant(Lx / 8)
y = Constant((Lx / 2, Ly / 2))
h_0.project(h + δh * max_value(0, 1 - inner(x - y, x - y) / r**2))
bcs = {
'thickness_in': h_in,
'momentum_in': as_vector((q_in, 0)),
'inflow_ids': (1, ),
'outflow_ids': (2, )
}
equation = plumes.models.shallow_water.make_equation(g, b, **bcs)
integrator = scheme(equation, z_0)
for step in range(num_steps):
integrator.step(dt)
h_n, q_n = integrator.state.split()
errors[k] = assemble(abs(h_n - h) * dx) / assemble(h * dx)
slope, intercept = np.polyfit(np.log2(1 / num_points), np.log2(errors), 1)
print(f'log(error) ~= {slope:5.3f} * log(dx) {intercept:+5.3f}')
assert slope > degree - 0.05
Vvec = fd.VectorFunctionSpace(mesh, "CG", 1)
# We'll also need the test and trial functions corresponding to this
# function space::
u = fd.TrialFunction(V)
v = fd.TestFunction(V)
# We declare a function over our function space and give it the
# value of our right hand side function::
# Permeability tensor
print("START: Read in reservoir fields")
Delta = np.array([Delta_x, Delta_y])
coords = fd.project(mesh.coordinates, Vvec).dat.data
kx_array = np.load('./spe10/spe10_kx.npy')
ky_array = np.load('./spe10/spe10_ky.npy')
kz_array = np.load('./spe10/spe10_kz.npy')
phi_array = np.load('./spe10/spe10_po.npy')
Kx = fd.Function(V)
Ky = fd.Function(V)
Kz = fd.Function(V)
phi = fd.Constat(0.2)
coords2ijk = np.vectorize(coords2ijk, excluded=['data_array', 'Delta'])
Kx.dat.data[...] = coords2ijk(coords[:, 0], coords[:, 1],
coords[:, 2], Delta=Delta, data_array=kx_array)
Ky.dat.data[...] = coords2ijk(coords[:, 0], coords[:, 1],
def test_3D_dirichlet_regions():
# Can't do mesh refinement because MeshHierarchy ruins the domain tags
mesh = Mesh("./3D_mesh.msh")
dim = mesh.geometric_dimension()
x = SpatialCoordinate(mesh)
solver_parameters_amg = {
"ksp_type": "cg",
"ksp_converged_reason": None,
"ksp_rtol": 1e-7,
"pc_type": "hypre",
"pc_hypre_type": "boomeramg",
"pc_hypre_boomeramg_max_iter": 5,
"pc_hypre_boomeramg_coarsen_type": "PMIS",
"pc_hypre_boomeramg_agg_nl": 2,
"pc_hypre_boomeramg_strong_threshold": 0.95,
"pc_hypre_boomeramg_interp_type": "ext+i",
"pc_hypre_boomeramg_P_max": 2,
"pc_hypre_boomeramg_relax_type_all": "sequential-Gauss-Seidel",
"pc_hypre_boomeramg_grid_sweeps_all": 1,
"pc_hypre_boomeramg_truncfactor": 0.3,
"pc_hypre_boomeramg_max_levels": 6,
}
S = VectorFunctionSpace(mesh, "CG", 1)
beta = 1.0
reg_solver = RegularizationSolver(
S,
mesh,
beta=beta,
gamma=0.0,
dx=dx,
design_domain=0,
solver_parameters=solver_parameters_amg,
)
# Exact solution with free Neumann boundary conditions for this domain
u_exact = Function(S)
u_exact_component = (
(-cos(x[0] * pi * 2) + 1)
* (-cos(x[1] * pi * 2) + 1)
* (-cos(x[2] * pi * 2) + 1)
)
u_exact.interpolate(
as_vector(tuple(u_exact_component for _ in range(dim)))
)
f = Function(S)
f_component = (
-beta
* (
4.0
* pi
* pi
* cos(2 * pi * x[0])
* (-cos(2 * pi * x[1]) + 1)
* (-cos(2 * pi * x[2]) + 1)
+ 4.0
* pi
* pi
* cos(2 * pi * x[1])
* (-cos(2 * pi * x[0]) + 1)
* (-cos(2 * pi * x[2]) + 1)
+ 4.0
* pi
* pi
* cos(2 * pi * x[2])
* (-cos(2 * pi * x[1]) + 1)
* (-cos(2 * pi * x[0]) + 1)
)
+ u_exact_component
)
f.interpolate(as_vector(tuple(f_component for _ in range(dim))))
theta = TestFunction(S)
rhs_form = inner(f, theta) * dx
velocity = Function(S)
rhs = assemble(rhs_form)
reg_solver.solve(velocity, rhs)
error = norm(
project(
domainify(u_exact, x) - velocity,
S,
solver_parameters=solver_parameters_amg,
)
)
assert error < 5e-2
- fd.conditional(fd.dot(u0, n) < 0, r *
fd.dot(u0, n)*Fw(s_in), 0.0)*fd.ds
- (r('+') - r('-'))*(un('+')*Fw(s_mid)('+') -
un('-')*Fw(s_mid)('-'))*fd.dS)
# Res.
R = F_p + F_s
prob = fd.NonlinearVariationalProblem(R, U, bcs=[bc0, bc1, bc3])
solver2ph = fd.NonlinearVariationalSolver(prob)
# %%
# 4) Solve problem
# initialize variables
xv, xp, xs = U.split()
vel = fd.project(u0, P1)
xp.rename('pressure')
xs.rename('saturation')
vel.rename('velocity')
outfile = fd.File(oFile)
it = 0
t = 0.0
while t < sim_time:
t += dt
print("*iteration= {:3d}, dtime= {:8.6f}, time={:8.6f}".format(it, dt, t))
solver2ph.solve()
# update solution
U0.assign(U)
# function space::
u = fd.TrialFunction(V)
v = fd.TestFunction(V)
# We declare a function over our function space and give it the
# value of our right hand side function::
# Permeability tensor
print("START: Read in reservoir fields")
Delta = np.array([Delta_x, Delta_y, Delta_z])
ct = 79.08e-11 # (1.0+0.2*3+0.8*4.947)*14.2 * 10**-6 kgf/cm2
mu = 0.003 # Pa-s
# coords = fd.project(mesh.coordinates, C).dat.data
coords = fd.project(mesh.coordinates, Vvec).dat.data
Kx = fd.interpolate(fd.Constant(98.7e-15), V)
Ky = fd.interpolate(fd.Constant(98.7e-15), V)
Kz = fd.interpolate(fd.Constant(98.7e-16), V)
phi = fd.interpolate(fd.Constant(0.25), V)
# well location
xw = np.array([118.872, 181.356, Lz / 2])
Tx = fd.Function(V)
Ty = fd.Function(V)
Tz = fd.Function(V)
w = fd.Function(V)
I = fd.Function(V)
def getFormStage(F,
butch,
u0,
t,
dt,
bcs=None,
splitting=None,
nullspace=None):
"""Given a time-dependent variational form and a
:class:`ButcherTableau`, produce UFL for the s-stage RK method.
:arg F: UFL form for the semidiscrete ODE/DAE
:arg butch: the :class:`ButcherTableau` for the RK method being used to
advance in time.
:arg u0: a :class:`Function` referring to the state of
the PDE system at time `t`
:arg t: a :class:`Constant` referring to the current time level.
Any explicit time-dependence in F is included
:arg dt: a :class:`Constant` referring to the size of the current
time step.
:arg splitting: a callable that maps the (floating point) Butcher matrix
a to a pair of matrices `A1, A2` such that `butch.A = A1 A2`. This is used
to vary between the classical RK formulation and Butcher's reformulation
that leads to a denser mass matrix with block-diagonal stiffness.
Only `AI` and `IA` are currently supported.
:arg bcs: optionally, a :class:`DirichletBC` object (or iterable thereof)
containing (possible time-dependent) boundary conditions imposed
on the system.
:arg nullspace: A list of tuples of the form (index, VSB) where
index is an index into the function space associated with `u`
and VSB is a :class: `firedrake.VectorSpaceBasis` instance to
be passed to a `firedrake.MixedVectorSpaceBasis` over the
larger space associated with the Runge-Kutta method
On output, we return a tuple consisting of several parts:
- Fnew, the :class:`Form`
- possibly a 4-tuple containing information needed to solve a mass matrix to update
the solution (this is empty for RadauIIA methods for which there is a trivial
update function.
- UU, the :class:`firedrake.Function` holding all the stage time values.
It lives in a :class:`firedrake.FunctionSpace` corresponding to the
s-way tensor product of the space on which the semidiscrete
form lives.
- `bcnew`, a list of :class:`firedrake.DirichletBC` objects to be posed
on the stages,
- 'nspnew', the :class:`firedrake.MixedVectorSpaceBasis` object
that represents the nullspace of the coupled system
- `gblah`, a list of tuples of the form (f, expr, method),
where f is a :class:`firedrake.Function` and expr is a
:class:`ufl.Expr`. At each time step, each expr needs to be
re-interpolated/projected onto the corresponding f in order
for Firedrake to pick up that time-dependent boundary
conditions need to be re-applied. The
interpolation/projection is encoded in method, which is
either `f.interpolate(expr-c*u0)` or `f.project(expr-c*u0)`, depending
on whether the function space for f supports interpolation or
not.
"""
v = F.arguments()[0]
V = v.function_space()
assert V == u0.function_space()
num_stages = butch.num_stages
num_fields = len(V)
# s-way product space for the stage variables
Vbig = reduce(mul, (V for _ in range(num_stages)))
VV = TestFunction(Vbig)
UU = Function(Vbig)
vecconst = np.vectorize(Constant)
C = vecconst(butch.c)
A = vecconst(butch.A)
# set up the pieces we need to work with to do our substitutions
nsxnf = (num_stages, num_fields)
if num_fields == 1:
u0bits = np.array([u0], dtype="O")
vbits = np.array([v], dtype="O")
if num_stages == 1: # single-stage method
VVbits = np.array([[VV]], dtype="O")
UUbits = np.array([[UU]], dtype="O")
else: # multi-stage methods
VVbits = np.reshape(split(VV), nsxnf)
UUbits = np.reshape(split(UU), nsxnf)
else:
u0bits = np.array(list(split(u0)), dtype="O")
vbits = np.array(list(split(v)), dtype="O")
VVbits = np.reshape(split(VV), nsxnf)
UUbits = np.reshape(split(UU), nsxnf)
split_form = extract_terms(F)
Fnew = Zero()
# first, process terms with a time derivative. I'm
# assuming we have something of the form inner(Dt(g(u0)), v)*dx
# For each stage i, this gets replaced with
# inner((g(stages[i]) - g(u0))/dt, v)*dx
# but we have to carefully handle the cases where g indexes into
# pieces of u
dtless = strip_dt_form(split_form.time)
if splitting is None or splitting == AI:
for i in range(num_stages):
repl = {t: t + C[i] * dt}
for j in range(num_fields):
repl[u0bits[j]] = UUbits[i][j] - u0bits[j]
repl[vbits[j]] = VVbits[i][j]
# Also get replacements right for indexing.
for j in range(num_fields):
for ii in np.ndindex(u0bits[j].ufl_shape):
repl[u0bits[j][ii]] = UUbits[i][j][ii] - u0bits[j][ii]
repl[vbits[j][ii]] = VVbits[i][j][ii]
Fnew += replace(dtless, repl)
# Now for the non-time derivative parts
for i in range(num_stages):
# replace test function
repl = {}
for k in range(num_fields):
repl[vbits[k]] = VVbits[i][k]
for ii in np.ndindex(vbits[k].ufl_shape):
repl[vbits[k][ii]] = VVbits[i][k][ii]
Ftmp = replace(split_form.remainder, repl)
# replace the solution with stage values
for j in range(num_stages):
repl = {t: t + C[j] * dt}
for k in range(num_fields):
repl[u0bits[k]] = UUbits[j][k]
for ii in np.ndindex(u0bits[k].ufl_shape):
repl[u0bits[k][ii]] = UUbits[j][k][ii]
# and sum the contribution
Fnew += A[i, j] * dt * replace(Ftmp, repl)
elif splitting == IA:
Ainv = np.vectorize(Constant)(np.linalg.inv(butch.A))
# time derivative part gets inverse of Butcher matrix.
for i in range(num_stages):
repl = {}
for k in range(num_fields):
repl[vbits[k]] = VVbits[i][k]
for ii in np.ndindex(vbits[k].ufl_shape):
repl[vbits[k][ii]] = VVbits[i][k][ii]
Ftmp = replace(dtless, repl)
for j in range(num_stages):
repl = {t: t + C[j] * dt}
for k in range(num_fields):
repl[u0bits[k]] = (UUbits[j][k] - u0bits[k])
for ii in np.ndindex(u0bits[k].ufl_shape):
repl[u0bits[k][ii]] = UUbits[j][k][ii] - u0bits[k][ii]
Fnew += Ainv[i, j] * replace(Ftmp, repl)
# rest of the operator: just diagonal!
for i in range(num_stages):
repl = {t: t + C[i] * dt}
for j in range(num_fields):
repl[u0bits[j]] = UUbits[i][j]
repl[vbits[j]] = VVbits[i][j]
# Also get replacements right for indexing.
for j in range(num_fields):
for ii in np.ndindex(u0bits[j].ufl_shape):
repl[u0bits[j][ii]] = UUbits[i][j][ii]
repl[vbits[j][ii]] = VVbits[i][j][ii]
Fnew += dt * replace(split_form.remainder, repl)
else:
raise NotImplementedError("Can't handle that splitting type")
if bcs is None:
bcs = []
bcsnew = []
gblah = []
# For each BC, we need a new BC for each stage
# so we need to figure out how the function is indexed (mixed + vec)
# and then set it to have the value of the original argument at
# time t+C[i]*dt.
for bc in bcs:
if num_fields == 1: # not mixed space
comp = bc.function_space().component
if comp is not None: # check for sub-piece of vector-valued
Vsp = V.sub(comp)
Vbigi = lambda i: Vbig[i].sub(comp)
else:
Vsp = V
Vbigi = lambda i: Vbig[i]
else: # mixed space
sub = bc.function_space_index()
comp = bc.function_space().component
if comp is not None: # check for sub-piece of vector-valued
Vsp = V.sub(sub).sub(comp)
Vbigi = lambda i: Vbig[sub + num_fields * i].sub(comp)
else:
Vsp = V.sub(sub)
Vbigi = lambda i: Vbig[sub + num_fields * i]
bcarg = bc._original_arg
for i in range(num_stages):
try:
gdat = interpolate(bcarg, Vsp)
gmethod = lambda gd, gc: gd.interpolate(gc)
except: # noqa: E722
gdat = project(bcarg, Vsp)
gmethod = lambda gd, gc: gd.project(gc)
gcur = replace(bcarg, {t: t + C[i] * dt})
bcsnew.append(DirichletBC(Vbigi(i), gdat, bc.sub_domain))
gblah.append((gdat, gcur, gmethod))
nspacenew = getNullspace(V, Vbig, butch, nullspace)
# For RIIA, we have an optimized update rule and don't need to
# build the variational form for doing updates.
# But something's broken with null spaces, so that's a TO-DO.
unew = Function(V)
Fupdate = inner(unew - u0, v) * dx
B = vectorize(Constant)(butch.b)
C = vectorize(Constant)(butch.c)
for i in range(num_stages):
repl = {t: t + C[i] * dt}
for k in range(num_fields):
repl[u0bits[k]] = UUbits[i][k]
for ii in np.ndindex(u0bits[k].ufl_shape):
repl[u0bits[k][ii]] = UUbits[i][k][ii]
eFFi = replace(split_form.remainder, repl)
Fupdate += dt * B[i] * eFFi
# And the BC's for the update -- just the original BC at t+dt
update_bcs = []
update_bcs_gblah = []
for bc in bcs:
if num_fields == 1: # not mixed space
comp = bc.function_space().component
if comp is not None: # check for sub-piece of vector-valued
Vsp = V.sub(comp)
else:
Vsp = V
else: # mixed space
sub = bc.function_space_index()
comp = bc.function_space().component
if comp is not None: # check for sub-piece of vector-valued
Vsp = V.sub(sub).sub(comp)
else:
Vsp = V.sub(sub)
bcarg = bc._original_arg
try:
gdat = interpolate(bcarg, Vsp)
gmethod = lambda gd, gc: gd.interpolate(gc)
except: # noqa: E722
gdat = project(bcarg, Vsp)
gmethod = lambda gd, gc: gd.project(gc)
gcur = replace(bcarg, {t: t + dt})
update_bcs.append(DirichletBC(Vsp, gdat, bc.sub_domain))
update_bcs_gblah.append((gdat, gcur, gmethod))
return (Fnew, (unew, Fupdate, update_bcs, update_bcs_gblah), UU, bcsnew,
gblah, nspacenew)