def __init__(self,
*args,
rayleigh_number=7.e5,
prandtl_number=0.0216,
stefan_number=0.046,
liquidus_temperature=0.,
hotwall_temperature=1.,
initial_temperature=-0.1546,
cutoff_length=0.5,
element_degrees=(1, 2, 2),
mesh_dimensions=(20, 40),
**kwargs):
if "solution" not in kwargs:
mesh = fe.RectangleMesh(nx=mesh_dimensions[0],
ny=mesh_dimensions[1],
Lx=cutoff_length,
Ly=1.)
element = sapphire.simulations.navier_stokes_boussinesq.element(
cell=mesh.ufl_cell(), degrees=element_degrees)
kwargs["solution"] = fe.Function(fe.FunctionSpace(mesh, element))
super().__init__(*args,
reynolds_number=1. / prandtl_number,
rayleigh_number=rayleigh_number,
prandtl_number=prandtl_number,
stefan_number=stefan_number,
liquidus_temperature=liquidus_temperature,
hotwall_temperature=hotwall_temperature,
initial_temperature=initial_temperature,
**kwargs)
def __init__(self, *args,
rayleigh_number = 7.e5,
prandtl_number = 0.0216,
stefan_number = 0.046,
liquidus_temperature = 0.,
hotwall_temperature = 1.,
initial_temperature = -0.1546,
cutoff_length = 0.5,
taylor_hood_pressure_degree = 1,
temperature_degree = 2,
mesh_dimensions = (20, 40),
**kwargs):
if "solution" not in kwargs:
kwargs["mesh"] = fe.RectangleMesh(
nx = mesh_dimensions[0],
ny = mesh_dimensions[1],
Lx = cutoff_length,
Ly = 1.)
super().__init__(
*args,
reynolds_number = 1./prandtl_number,
rayleigh_number = rayleigh_number,
prandtl_number = prandtl_number,
stefan_number = stefan_number,
liquidus_temperature = liquidus_temperature,
hotwall_temperature = hotwall_temperature,
initial_temperature = initial_temperature,
**kwargs)
def test_diagnostic_solver_side_friction():
model = icepack.models.IceShelf()
opts = {"dirichlet_ids": [1], "side_wall_ids": [3, 4]}
mesh = firedrake.RectangleMesh(32, 32, Lx, Ly)
degree = 2
V = firedrake.VectorFunctionSpace(mesh, "CG", degree)
Q = firedrake.FunctionSpace(mesh, "CG", degree)
x, y = firedrake.SpatialCoordinate(mesh)
u_initial = interpolate(as_vector((exact_u(x), 0)), V)
h = interpolate(h0 - dh * x / Lx, Q)
A = interpolate(firedrake.Constant(icepack.rate_factor(T)), Q)
# Choose the side wall friction coefficient so that, assuming the ice is
# sliding at the maximum speed for the solution without friction, the
# stress is 10 kPa.
from icepack.constants import weertman_sliding_law as m
τ = 0.01
u_max = icepack.norm(u_initial, norm_type="Linfty")
Cs = firedrake.Constant(τ * u_max**(-1 / m))
solver = icepack.solvers.FlowSolver(model, **opts)
fields = {
"velocity": u_initial,
"thickness": h,
"fluidity": A,
"side_friction": Cs
}
u = solver.diagnostic_solve(**fields)
assert icepack.norm(u) < icepack.norm(u_initial)
def test_diagnostic_solver_convergence():
# Create an ice shelf model
ice_shelf = icepack.models.IceShelf()
opts = {'dirichlet_ids': [1], 'side_wall_ids': [3, 4], 'tol': 1e-12}
# Solve the ice shelf model for successively higher mesh resolution
for degree in range(1, 4):
delta_x, error = [], []
for N in range(16, 97 - 32 * (degree - 1), 4):
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
V = firedrake.VectorFunctionSpace(mesh, 'CG', degree)
Q = firedrake.FunctionSpace(mesh, 'CG', degree)
u_exact = interpolate(as_vector((exact_u(x), 0)), V)
u_guess = interpolate(u_exact + as_vector((perturb_u(x, y), 0)), V)
h = interpolate(h0 - dh * x / Lx, Q)
A = interpolate(firedrake.Constant(icepack.rate_factor(T)), Q)
u = ice_shelf.diagnostic_solve(h=h, A=A, u0=u_guess, **opts)
error.append(norm(u_exact - u) / norm(u_exact))
delta_x.append(Lx / N)
print(delta_x[-1], error[-1])
# Fit the error curve and check that the convergence rate is what we
# expect
log_delta_x = np.log2(np.array(delta_x))
log_error = np.log2(np.array(error))
slope, intercept = np.polyfit(log_delta_x, log_error, 1)
print('log(error) ~= {:g} * log(dx) + {:g}'.format(slope, intercept))
assert slope > degree + 0.8
def test_hybrid_prognostic_solve():
Lx, Ly = 20e3, 20e3
h0, dh = 500.0, 100.0
T = 254.15
u_in = 100.0
model = icepack.models.HybridModel()
opts = {'dirichlet_ids': [1], 'side_wall_ids': [3, 4], 'tol': 1e-12}
Nx, Ny = 32, 32
mesh2d = firedrake.RectangleMesh(Nx, Ny, Lx, Ly)
mesh = firedrake.ExtrudedMesh(mesh2d, layers=1)
V = firedrake.VectorFunctionSpace(mesh,
dim=2,
family='CG',
degree=2,
vfamily='GL',
vdegree=1)
Q = firedrake.FunctionSpace(mesh,
family='CG',
degree=2,
vfamily='DG',
vdegree=0)
x, y, ζ = firedrake.SpatialCoordinate(mesh)
height_above_flotation = 10.0
d = -ρ_I / ρ_W * (h0 - dh) + height_above_flotation
ρ = ρ_I - ρ_W * d**2 / (h0 - dh)**2
Z = icepack.rate_factor(T) * (ρ * g * h0 / 4)**n
q = 1 - (1 - (dh / h0) * (x / Lx))**(n + 1)
ux = u_in + Z * q * Lx * (h0 / dh) / (n + 1)
u0 = interpolate(firedrake.as_vector((ux, 0)), V)
thickness = h0 - dh * x / Lx
β = 1 / 2
α = β * ρ / ρ_I * dh / Lx
h = interpolate(h0 - dh * x / Lx, Q)
h_inflow = h.copy(deepcopy=True)
ds = (1 + β) * ρ / ρ_I * dh
s = interpolate(d + h0 - dh + ds * (1 - x / Lx), Q)
b = interpolate(s - h, Q)
C = interpolate(α * (ρ_I * g * thickness) * ux**(-1 / m), Q)
A = firedrake.Constant(icepack.rate_factor(T))
final_time, dt = 1.0, 1.0 / 12
num_timesteps = int(final_time / dt)
u = model.diagnostic_solve(u0=u0, h=h, s=s, C=C, A=A, **opts)
a0 = firedrake.Constant(0)
a = interpolate((model.prognostic_solve(dt, h0=h, a=a0, u=u) - h) / dt, Q)
for k in range(num_timesteps):
h = model.prognostic_solve(dt, h0=h, a=a, u=u, h_inflow=h_inflow)
s = icepack.compute_surface(h=h, b=b)
u = model.diagnostic_solve(u0=u, h=h, s=s, C=C, A=A, **opts)
assert icepack.norm(h, norm_type='Linfty') < np.inf
def test_plot_mesh():
nx, ny = 32, 32
Lx, Ly = 1e5, 1e5
mesh = firedrake.RectangleMesh(nx, ny, Lx, Ly)
fig, axes = icepack.plot.subplots()
icepack.plot.triplot(mesh, axes=axes)
legend = axes.legend()
assert legend is not None
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 test_diagnostic_solver_parameterization():
# Define a new viscosity functional, parameterized in terms of the
# rheology `B` instead of the fluidity `A`
from firedrake import inner, grad, sym, tr as trace, Identity, sqrt
def M(ε, B):
I = Identity(2)
tr_ε = trace(ε)
ε_e = sqrt((inner(ε, ε) + tr_ε**2) / 2)
μ = 0.5 * B * ε_e**(1/n - 1)
return 2 * μ * (ε + tr_ε * I)
def ε(u):
return sym(grad(u))
def viscosity(**kwargs):
u = kwargs['velocity']
h = kwargs['thickness']
B = kwargs['rheology']
return n/(n + 1) * h * inner(M(ε(u), B), ε(u))
# Make a model object with our new viscosity functional
model = icepack.models.IceShelf(viscosity=viscosity)
opts = {'dirichlet_ids': [1, 3, 4]}
# Same as before
delta_x, error = [], []
for N in range(16, 65, 4):
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
degree = 2
V = firedrake.VectorFunctionSpace(mesh, 'CG', degree)
Q = firedrake.FunctionSpace(mesh, 'CG', degree)
u_exact = interpolate(as_vector((exact_u(x), 0)), V)
u_guess = interpolate(as_vector((exact_u(x) + perturb_u(x, y), 0)), V)
h = interpolate(h0 - dh * x / Lx, Q)
B = interpolate(firedrake.Constant(icepack.rate_factor(T)**(-1/n)), Q)
solver = icepack.solvers.FlowSolver(model, **opts)
u = solver.diagnostic_solve(
velocity=u_guess,
thickness=h,
rheology=B
)
error.append(norm(u_exact - u) / norm(u_exact))
delta_x.append(Lx / N)
print(delta_x[-1], error[-1])
log_delta_x = np.log2(np.array(delta_x))
log_error = np.log2(np.array(error))
slope, intercept = np.polyfit(log_delta_x, log_error, 1)
print('log(error) ~= {:g} * log(dx) + {:g}'.format(slope, intercept))
assert slope > degree - 0.05
def test_order_0():
def h_expr(x):
return h0 - dh * x / Lx
def s_expr(x):
return d + h0 - dh + ds * (1 - x / Lx)
A = firedrake.Constant(icepack.rate_factor(254.15))
C = firedrake.Constant(0.001)
opts = {'dirichlet_ids': [1], 'side_wall_ids': [3, 4], 'tolerance': 1e-14}
Nx, Ny = 64, 64
mesh2d = firedrake.RectangleMesh(Nx, Ny, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh2d)
Q2d = firedrake.FunctionSpace(mesh2d, family='CG', degree=2)
V2d = firedrake.VectorFunctionSpace(mesh2d, family='CG', degree=2)
h = firedrake.interpolate(h_expr(x), Q2d)
s = firedrake.interpolate(s_expr(x), Q2d)
u_expr = firedrake.as_vector((exact_u(x), 0))
u0 = firedrake.interpolate(u_expr, V2d)
model2d = icepack.models.IceStream()
solver2d = icepack.solvers.FlowSolver(model2d, **opts)
u2d = solver2d.diagnostic_solve(velocity=u0,
thickness=h,
surface=s,
fluidity=A,
friction=C)
mesh = firedrake.ExtrudedMesh(mesh2d, layers=1)
x, y, ζ = firedrake.SpatialCoordinate(mesh)
Q3d = firedrake.FunctionSpace(mesh,
family='CG',
degree=2,
vfamily='DG',
vdegree=0)
V3d = firedrake.VectorFunctionSpace(mesh,
dim=2,
family='CG',
degree=2,
vfamily='GL',
vdegree=0)
h = firedrake.interpolate(h_expr(x), Q3d)
s = firedrake.interpolate(s_expr(x), Q3d)
u_expr = firedrake.as_vector((exact_u(x), 0))
u0 = firedrake.interpolate(u_expr, V3d)
model3d = icepack.models.HybridModel()
solver3d = icepack.solvers.FlowSolver(model3d, **opts)
u3d = solver3d.diagnostic_solve(velocity=u0,
thickness=h,
surface=s,
fluidity=A,
friction=C)
U2D, U3D = u2d.dat.data_ro, u3d.dat.data_ro
assert np.linalg.norm(U3D - U2D) / np.linalg.norm(U2D) < 1e-2
def test_max_dim():
mesh = fd.UnitSquareMesh(100, 100)
assert pytest.approx(max_mesh_dimension(mesh), 1.0)
mesh = fd.RectangleMesh(10, 20, 5.0, 3.0)
assert pytest.approx(max_mesh_dimension(mesh), 5.0)
mesh = fd.CubeMesh(10, 10, 10, 8.0)
assert pytest.approx(max_mesh_dimension(mesh), 8.0)
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 test_ice_shelf_prognostic_solver(solver_type):
ρ = ρ_I * (1 - ρ_I / ρ_W)
Lx, Ly = 20.0e3, 20.0e3
h0 = 500.0
u0 = 100.0
T = 254.15
model = icepack.models.IceShelf(mass_transport=solver_type())
opts = {'dirichlet_ids': [1], 'side_wall_ids': [3, 4], 'tol': 1e-12}
delta_x, error = [], []
for N in range(16, 65, 4):
delta_x.append(Lx / N)
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
degree = 2
V = firedrake.VectorFunctionSpace(mesh, 'CG', degree)
Q = firedrake.FunctionSpace(mesh, 'CG', degree)
q = (n + 1) * (ρ * g * h0 * u0 / 4)**n * icepack.rate_factor(T)
ux = (u0**(n + 1) + q * x)**(1 / (n + 1))
h = interpolate(h0 * u0 / ux, Q)
h_initial = h.copy(deepcopy=True)
A = firedrake.Constant(icepack.rate_factor(T))
a = firedrake.Constant(0)
u_guess = interpolate(firedrake.as_vector((ux, 0)), V)
u = model.diagnostic_solve(u0=u_guess, h=h, A=A, **opts)
final_time, dt = 1.0, 1.0 / 12
num_timesteps = int(final_time / dt)
for k in range(num_timesteps):
h = model.prognostic_solve(dt, h0=h, a=a, u=u, h_inflow=h_initial)
u = model.diagnostic_solve(u0=u, h=h, A=A, **opts)
error.append(norm(h - h_initial) / norm(h_initial))
print(delta_x[-1], error[-1])
log_delta_x = np.log2(np.array(delta_x))
log_error = np.log2(np.array(error))
slope, intercept = np.polyfit(log_delta_x, log_error, 1)
print('log(error) ~= {:g} * log(dx) + {:g}'.format(slope, intercept))
assert slope > degree - 0.05
def test_computing_surface():
N = 16
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
degree = 2
Q = firedrake.FunctionSpace(mesh, "CG", degree)
x, y = firedrake.SpatialCoordinate(mesh)
h = interpolate(h0 - dh * x / Lx, Q)
b0 = ρ_I / ρ_W * (dh / 2 - h0)
b = interpolate(firedrake.Constant(b0), Q)
s = icepack.compute_surface(thickness=h, bed=b)
x0, y0 = Lx / 2, Ly / 2
assert abs(s((x0, y0)) - (1 - ρ_I / ρ_W) * h((x0, y0))) < 1e-8
def test_mass_transport_solver_convergence(solver_type):
Lx, Ly = 1.0, 1.0
u0 = 1.0
h_in, dh = 1.0, 0.2
delta_x, error = [], []
model = icepack.models.IceShelf()
for N in range(24, 97, 4):
delta_x.append(Lx / N)
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
degree = 1
V = firedrake.VectorFunctionSpace(mesh, family='CG', degree=degree)
Q = firedrake.FunctionSpace(mesh, family='CG', degree=degree)
solver = icepack.solvers.FlowSolver(
model, prognostic_solver_type=solver_type
)
h0 = interpolate(h_in - dh * x / Lx, Q)
a = firedrake.Function(Q)
u = interpolate(firedrake.as_vector((u0, 0)), V)
T = 0.5
δx = 1.0 / N
δt = δx / u0
num_timesteps = int(T / δt)
h = h0.copy(deepcopy=True)
for step in range(num_timesteps):
h = solver.prognostic_solve(
δt,
thickness=h,
velocity=u,
accumulation=a,
thickness_inflow=h0
)
z = x - u0 * num_timesteps * δt
h_exact = interpolate(h_in - dh/Lx * firedrake.max_value(0, z), Q)
error.append(norm(h - h_exact) / norm(h_exact))
print(delta_x[-1], error[-1])
log_delta_x = np.log2(np.array(delta_x))
log_error = np.log2(np.array(error))
slope, intercept = np.polyfit(log_delta_x, log_error, 1)
print('log(error) ~= {:g} * log(dx) + {:g}'.format(slope, intercept))
assert slope > degree - 0.1
def test_vertical_velocity():
Lx, Ly = 20e3, 20e3
nx, ny = 48, 48
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)
Q3D = firedrake.FunctionSpace(mesh,
family="DG",
degree=2,
vfamily="GL",
vdegree=6)
V = firedrake.VectorFunctionSpace(mesh,
dim=2,
family="CG",
degree=2,
vfamily="GL",
vdegree=5)
# note we should call the families in the vertical velocity solver.
x, y, ζ = firedrake.SpatialCoordinate(mesh)
u_inflow = 1.0
v_inflow = 2.0
mu = 0.003
mv = 0.001
b = firedrake.interpolate(firedrake.Constant(0.0), Q)
s = firedrake.interpolate(firedrake.Constant(1000.0), Q)
h = firedrake.interpolate(s - b, Q)
u = firedrake.interpolate(
firedrake.as_vector((mu * x + u_inflow, mv * y + v_inflow)), V)
m = -0.01
def analytic_vertical_velocity(h, ζ, mu, mv, m, Q3D):
return firedrake.interpolate(
firedrake.Constant(m) - (firedrake.Constant(mu + mv) * h * ζ), Q3D)
w = firedrake.interpolate(
icepack.utilities.vertical_velocity(u, h, m=m) * h, Q3D)
w_analytic = analytic_vertical_velocity(h, ζ, mu, mv, m, Q3D)
assert np.mean(np.abs(w.dat.data - w_analytic.dat.data)) < 10e-9
def basemesh(L, mx, my=-1, quadrilateral=False):
'''Set up base mesh of intervals on [-L,L] if my<0. For 2D base mesh
(my>0) use triangles, or optionally quadilaterals, on [-L,L]x[-L,L].'''
if my > 0:
base_mesh = fd.RectangleMesh(mx,
my,
2.0 * L,
2.0 * L,
quadrilateral=quadrilateral)
base_mesh.coordinates.dat.data[:, 0] -= L
base_mesh.coordinates.dat.data[:, 1] -= L
else:
base_mesh = fd.IntervalMesh(mx, length_or_left=0.0, right=2.0 * L)
base_mesh.coordinates.dat.data[:] -= L
return base_mesh
def test_diagnostic_solver_convergence(solver_type):
# Create an ice shelf model
model = icepack.models.IceShelf()
opts = {
"dirichlet_ids": [1],
"side_wall_ids": [3, 4],
"diagnostic_solver_type": solver_type,
}
# Solve the ice shelf model for successively higher mesh resolution
for degree in range(1, 4):
delta_x, error = [], []
for N in range(16, 97 - 32 * (degree - 1), 4):
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
V = firedrake.VectorFunctionSpace(mesh, "CG", degree)
Q = firedrake.FunctionSpace(mesh, "CG", degree)
u_exact = interpolate(as_vector((exact_u(x), 0)), V)
u_guess = interpolate(u_exact + as_vector((perturb_u(x, y), 0)), V)
h = interpolate(h0 - dh * x / Lx, Q)
A = interpolate(firedrake.Constant(icepack.rate_factor(T)), Q)
solver = icepack.solvers.FlowSolver(model, **opts)
u = solver.diagnostic_solve(
velocity=u_guess,
thickness=h,
fluidity=A,
strain_rate_min=firedrake.Constant(0.0),
)
error.append(norm(u_exact - u) / norm(u_exact))
delta_x.append(Lx / N)
print(delta_x[-1], error[-1])
# Fit the error curve and check that the convergence rate is what we
# expect
log_delta_x = np.log2(np.array(delta_x))
log_error = np.log2(np.array(error))
slope, intercept = np.polyfit(log_delta_x, log_error, 1)
print(f"log(error) ~= {slope:g} * log(dx) + {intercept:g}")
assert slope > degree + 0.8
def test_diagnostic_solver_convergence():
model = icepack.models.IceStream()
opts = {"dirichlet_ids": [1], "side_wall_ids": [3, 4]}
for degree in range(1, 4):
delta_x, error = [], []
for N in range(16, 97 - 32 * (degree - 1), 4):
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
Q = firedrake.FunctionSpace(mesh, "CG", degree)
V = firedrake.VectorFunctionSpace(mesh, "CG", degree)
u_exact = interpolate(as_vector((exact_u(x), 0)), V)
u_guess = interpolate(u_exact + as_vector((perturb_u(x, y), 0)), V)
h = interpolate(h0 - dh * x / Lx, Q)
s = interpolate(d + h0 - dh + ds * (1 - x / Lx), Q)
C = interpolate(friction(x), Q)
A = interpolate(firedrake.Constant(icepack.rate_factor(T)), Q)
solver = icepack.solvers.FlowSolver(model, **opts)
u = solver.diagnostic_solve(
velocity=u_guess,
thickness=h,
surface=s,
fluidity=A,
friction=C,
strain_rate_min=firedrake.Constant(0.0),
)
error.append(norm(u_exact - u) / norm(u_exact))
delta_x.append(Lx / N)
print(delta_x[-1], error[-1])
log_delta_x = np.log2(np.array(delta_x))
log_error = np.log2(np.array(error))
slope, intercept = np.polyfit(log_delta_x, log_error, 1)
print(f"log(error) ~= {slope:g} * log(dx) + {intercept:g}")
assert slope > degree + 0.9
def test_mass_transport_solver_convergence():
Lx, Ly = 1.0, 1.0
u0 = 1.0
h_in, dh = 1.0, 0.2
delta_x, error = [], []
mass_transport = MassTransport()
for N in range(16, 97, 4):
delta_x.append(Lx / N)
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
degree = 1
V = firedrake.VectorFunctionSpace(mesh, family='CG', degree=degree)
Q = firedrake.FunctionSpace(mesh, family='CG', degree=degree)
h = interpolate(h_in - dh * x / Lx, Q)
a = firedrake.Function(Q)
u = interpolate(firedrake.as_vector((u0, 0)), V)
T = 0.5
num_timesteps = int(0.5 * N * u0 * T / Lx)
dt = T / num_timesteps
for k in range(num_timesteps):
h = mass_transport.solve(dt, h0=h, a=a, u=u)
z = x - u0 * T
h_exact = interpolate(h_in - dh / Lx * firedrake.max_value(0, z), Q)
error.append(norm(h - h_exact) / norm(h_exact))
print(delta_x[-1], error[-1])
log_delta_x = np.log2(np.array(delta_x))
log_error = np.log2(np.array(error))
slope, intercept = np.polyfit(log_delta_x, log_error, 1)
assert slope > degree - 0.05
print(slope, intercept)
def test_diagnostic_solver_convergence():
ice_stream = icepack.models.IceStream()
opts = {'dirichlet_ids': [1], 'side_wall_ids': [3, 4], 'tol': 1e-12}
for degree in range(1, 4):
delta_x, error = [], []
for N in range(16, 97 - 32 * (degree - 1), 4):
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
Q = firedrake.FunctionSpace(mesh, 'CG', degree)
V = firedrake.VectorFunctionSpace(mesh, 'CG', degree)
u_exact = interpolate(as_vector((exact_u(x), 0)), V)
u_guess = interpolate(u_exact + as_vector((perturb_u(x, y), 0)), V)
h = interpolate(h0 - dh * x / Lx, Q)
s = interpolate(d + h0 - dh + ds * (1 - x / Lx), Q)
C = interpolate(friction(x), Q)
A = interpolate(firedrake.Constant(icepack.rate_factor(T)), Q)
u = ice_stream.diagnostic_solve(h=h,
s=s,
A=A,
C=C,
u0=u_guess,
**opts)
error.append(norm(u_exact - u) / norm(u_exact))
delta_x.append(Lx / N)
print(delta_x[-1], error[-1])
log_delta_x = np.log2(np.array(delta_x))
log_error = np.log2(np.array(error))
slope, intercept = np.polyfit(log_delta_x, log_error, 1)
print('log(error) ~= {:g} * log(dx) + {:g}'.format(slope, intercept))
assert slope > degree + 0.9
def test_diagnostic_solver_side_friction():
ice_shelf = icepack.models.IceShelf()
opts = {'dirichlet_ids': [1], 'side_wall_ids': [3, 4], 'tol': 1e-12}
mesh = firedrake.RectangleMesh(32, 32, Lx, Ly)
degree = 2
V = firedrake.VectorFunctionSpace(mesh, 'CG', degree)
Q = firedrake.FunctionSpace(mesh, 'CG', degree)
x, y = firedrake.SpatialCoordinate(mesh)
u_initial = interpolate(as_vector((exact_u(x), 0)), V)
h = interpolate(h0 - dh * x / Lx, Q)
A = interpolate(firedrake.Constant(icepack.rate_factor(T)), Q)
# Choose the side wall friction coefficient so that, assuming the ice is
# sliding at the maximum speed for the solution without friction, the
# stress is 10 kPa.
from icepack.constants import weertman_sliding_law as m
τ = 0.01
u_max = icepack.norm(u_initial, norm_type='Linfty')
Cs = firedrake.Constant(τ * u_max**(-1 / m))
u = ice_shelf.diagnostic_solve(u0=u_initial, h=h, A=A, Cs=Cs, **opts)
assert icepack.norm(u) < icepack.norm(u_initial)
import os
import argparse
import firedrake
import icepack.plot
parser = argparse.ArgumentParser()
parser.add_argument('--input')
parser.add_argument('--level', type=int)
parser.add_argument('--output')
args = parser.parse_args()
Lx, Ly = 640e3, 80e3
ny = 20
nx = int(Lx / Ly) * ny
coarse_mesh = firedrake.RectangleMesh(nx, ny, Lx, Ly)
mesh_hierarchy = firedrake.MeshHierarchy(coarse_mesh, args.level)
mesh = mesh_hierarchy[args.level]
Q = firedrake.FunctionSpace(mesh, family='CG', degree=1)
V = firedrake.VectorFunctionSpace(mesh, family='CG', degree=1)
h = firedrake.Function(Q)
u = firedrake.Function(V)
input_name = os.path.splitext(args.input)[0]
with firedrake.DumbCheckpoint(input_name, mode=firedrake.FILE_READ) as chk:
timesteps, indices = chk.get_timesteps()
chk.set_timestep(timesteps[-1], idx=indices[-1])
chk.load(h, name='h')
# (at your option) any later version.
#
# The full text of the license can be found in the file LICENSE in the
# icepack source directory or at <http://www.gnu.org/licenses/>.
import numpy as np
import firedrake
from firedrake import assemble, inner, as_vector, Constant, dx, ds_t, ds_b
import icepack.models
from icepack.constants import (year, thermal_diffusivity as α,
melting_temperature as Tm)
# Using the same mesh and data for every test case.
Nx, Ny = 32, 32
Lx, Ly = 20e3, 20e3
mesh2d = firedrake.RectangleMesh(Nx, Ny, Lx, Ly)
mesh = firedrake.ExtrudedMesh(mesh2d, layers=1)
Q = firedrake.FunctionSpace(mesh,
family='CG',
degree=2,
vfamily='GL',
vdegree=4)
V = firedrake.VectorFunctionSpace(mesh,
dim=2,
family='CG',
degree=2,
vfamily='GL',
vdegree=4)
W = firedrake.FunctionSpace(mesh,
family='DG',
q0bar = (0.0, 0.0)
noflow = [TOP, BOTTOM]
# problem parameters
verbose = True
freq_res = 50
CFL = 0.25
sim_time = 50.0 # simulation time
# %%
# Process section (simulation)
# ============================
# 2) Define mesh
print('* define mesh')
mesh = fd.RectangleMesh(nx * n, ny * n, Lx, Ly, quadrilateral=quad_mesh)
if verbose:
fd.triplot(mesh)
plt.legend()
plt.savefig("plots/darcy_rt.png")
# ----
# 3) Setting problem (FunctionSpace, Init.Condition, VariationalForms)
print('* setting problem')
# 3.1) # Define function space for system
if quad_mesh:
RT = fd.FunctionSpace(mesh, "RTCF", order)
DG = fd.FunctionSpace(mesh, "DQ", order - 1)
tol = 1e-15
verbose = False
freq_res = 100
sim_time = 1e5 # simulation time
dtime = 3e3
cfl = 0.5
# %%
# Process section (simulation)
# -----------------------------
# 2) Define mesh
print('* define mesh')
mesh = fd.RectangleMesh(nx * refine,
ny * refine,
Lx,
Ly,
quadrilateral=quad_mesh)
if verbose:
fd.triplot(mesh)
plt.legend()
plt.savefig("plots/adr_dg_mesh.png")
# 3) Setting problem (FunctionSpace, Init.Bound.Condition, VariationalForms)
# 3.1) Set Function spaces
if quad_mesh:
DG1 = fd.FunctionSpace(mesh, "DQ", order)
vDG1 = fd.VectorFunctionSpace(mesh, "DQ", order)
Mh = fd.FunctionSpace(mesh, "HDiv Trace", order)
def test_hybrid_prognostic_solve(solver_type):
Lx, Ly = 20e3, 20e3
h0, dh = 500.0, 100.0
T = 254.15
u_in = 100.0
model = icepack.models.HybridModel()
opts = {
'dirichlet_ids': [1],
'side_wall_ids': [3, 4],
'prognostic_solver_type': solver_type
}
Nx, Ny = 32, 32
mesh2d = firedrake.RectangleMesh(Nx, Ny, Lx, Ly)
mesh = firedrake.ExtrudedMesh(mesh2d, layers=1)
V = firedrake.VectorFunctionSpace(
mesh, 'CG', 2, vfamily='GL', vdegree=1, dim=2
)
Q = firedrake.FunctionSpace(mesh, 'CG', 2, vfamily='DG', vdegree=0)
x, y, ζ = firedrake.SpatialCoordinate(mesh)
height_above_flotation = 10.0
d = -ρ_I / ρ_W * (h0 - dh) + height_above_flotation
ρ = ρ_I - ρ_W * d**2 / (h0 - dh)**2
Z = icepack.rate_factor(T) * (ρ * g * h0 / 4)**n
q = 1 - (1 - (dh/h0) * (x/Lx))**(n + 1)
ux = u_in + Z * q * Lx * (h0/dh) / (n + 1)
u0 = interpolate(firedrake.as_vector((ux, 0)), V)
thickness = h0 - dh * x / Lx
β = 1/2
α = β * ρ / ρ_I * dh / Lx
h = interpolate(h0 - dh * x / Lx, Q)
h_inflow = h.copy(deepcopy=True)
ds = (1 + β) * ρ / ρ_I * dh
s = interpolate(d + h0 - dh + ds * (1 - x / Lx), Q)
b = interpolate(s - h, Q)
C = interpolate(α * (ρ_I * g * thickness) * ux**(-1/m), Q)
A = firedrake.Constant(icepack.rate_factor(T))
final_time, dt = 1.0, 1.0/12
num_timesteps = int(final_time / dt)
solver = icepack.solvers.FlowSolver(model, **opts)
u = solver.diagnostic_solve(
velocity=u0, thickness=h, surface=s, fluidity=A, friction=C
)
a = firedrake.Function(Q)
h_n = solver.prognostic_solve(
dt, thickness=h, velocity=u, accumulation=a, thickness_inflow=h_inflow
)
a.interpolate((h_n - h) / dt)
for k in range(num_timesteps):
h = solver.prognostic_solve(
dt,
thickness=h,
velocity=u,
accumulation=a,
thickness_inflow=h_inflow
)
s = icepack.compute_surface(thickness=h, bed=b)
u = solver.diagnostic_solve(
velocity=u,
thickness=h,
surface=s,
fluidity=A,
friction=C
)
assert icepack.norm(h, norm_type='Linfty') < np.inf
tol = 1e-15
verbose = False
freq_res = 60
sim_time = 1e6 # simulation time
cfl = 0.125
tau_e = 1
# %%
# Process section (simulation)
# -----------------------------
# 2) Define mesh
print('* define mesh')
mesh = fd.RectangleMesh(int(nx * refine),
int(ny * refine),
Lx,
Ly,
quadrilateral=quad_mesh)
if verbose:
fd.triplot(mesh)
plt.legend()
plt.savefig("plots/adr_dg_mesh.png")
# 3) Setting problem (FunctionSpace, Init.Bound.Condition, VariationalForms)
# 3.1) Set Function spaces
if quad_mesh:
DG1 = fd.FunctionSpace(mesh, "DQ", order)
vDG1 = fd.VectorFunctionSpace(mesh, "DQ", order)
Mh = fd.FunctionSpace(mesh, "HDiv Trace", order)
def test_ice_shelf_inverse(solver_type):
import icepack
Nx, Ny = 32, 32
Lx, Ly = 20e3, 20e3
u0 = 100
h0, δh = 500, 100
T0, δT = 254.15, 5.0
A0 = icepack.rate_factor(T0)
δA = icepack.rate_factor(T0 + δT) - A0
def exact_u(x):
ρ = ρ_I * (1 - ρ_I / ρ_W)
Z = icepack.rate_factor(T0) * (ρ * g * h0 / 4)**n
q = 1 - (1 - (δh / h0) * (x / Lx))**(n + 1)
δu = Z * q * Lx * (h0 / δh) / (n + 1)
return u0 + δu
mesh = firedrake.RectangleMesh(Nx, Ny, Lx, Ly)
degree = 2
V = firedrake.VectorFunctionSpace(mesh, 'CG', degree)
Q = firedrake.FunctionSpace(mesh, 'CG', degree)
x, y = firedrake.SpatialCoordinate(mesh)
u_initial = interpolate(as_vector((exact_u(x), 0)), V)
q_initial = interpolate(firedrake.Constant(0), Q)
h = interpolate(h0 - δh * x / Lx, Q)
def viscosity(u, h, q):
A = A0 * firedrake.exp(-q / n)
return icepack.models.viscosity.viscosity_depth_averaged(u, h, A)
ice_shelf = icepack.models.IceShelf(viscosity=viscosity)
dirichlet_ids = [1, 3, 4]
tol = 1e-12
r = firedrake.sqrt((x / Lx - 1 / 2)**2 + (y / Ly - 1 / 2)**2)
R = 1 / 4
expr = firedrake.max_value(0, δA * (1 - (r / R)**2))
q_true = firedrake.interpolate(-n * firedrake.ln(1 + expr / A0), Q)
u_true = ice_shelf.diagnostic_solve(h=h,
q=q_true,
u0=u_initial,
dirichlet_ids=dirichlet_ids,
tol=tol)
area = firedrake.assemble(firedrake.Constant(1) * dx(mesh))
def callback(inverse_solver):
E, R = inverse_solver.objective, inverse_solver.regularization
misfit = firedrake.assemble(E) / area
regularization = firedrake.assemble(R) / area
q = inverse_solver.parameter
error = firedrake.norm(q - q_true) / np.sqrt(area)
print(misfit, regularization, error)
L = 1e-4 * Lx
problem = icepack.inverse.InverseProblem(
model=ice_shelf,
method=icepack.models.IceShelf.diagnostic_solve,
objective=lambda u: 0.5 * (u - u_true)**2 * dx,
regularization=lambda q: 0.5 * L**2 * inner(grad(q), grad(q)) * dx,
state_name='u',
state=u_initial,
parameter_name='q',
parameter=q_initial,
model_args={
'h': h,
'u0': u_initial,
'tol': tol
},
dirichlet_ids=dirichlet_ids)
solver = solver_type(problem, callback)
# Set an absolute tolerance so that we stop whenever the RMS velocity
# errors are less than 0.1 m/yr
area = firedrake.assemble(firedrake.Constant(1) * dx(mesh))
atol = 0.5 * 0.01 * area
max_iterations = 100
iterations = solver.solve(rtol=0, atol=atol, max_iterations=max_iterations)
print('Number of iterations: {}'.format(iterations))
assert iterations < max_iterations
q = solver.parameter
assert firedrake.norm(q - q_true) / firedrake.norm(q_initial -
q_true) < 1 / 4
def test_ice_shelf_inverse_with_noise(solver_type):
import icepack
Nx, Ny = 32, 32
Lx, Ly = 20e3, 20e3
u0 = 100
h0, δh = 500, 100
T0, δT = 254.15, 5.0
A0 = icepack.rate_factor(T0)
δA = icepack.rate_factor(T0 + δT) - A0
def exact_u(x):
ρ = ρ_I * (1 - ρ_I / ρ_W)
Z = icepack.rate_factor(T0) * (ρ * g * h0 / 4)**n
q = 1 - (1 - (δh / h0) * (x / Lx))**(n + 1)
δu = Z * q * Lx * (h0 / δh) / (n + 1)
return u0 + δu
mesh = firedrake.RectangleMesh(Nx, Ny, Lx, Ly)
degree = 2
V = firedrake.VectorFunctionSpace(mesh, 'CG', degree)
Q = firedrake.FunctionSpace(mesh, 'CG', degree)
x, y = firedrake.SpatialCoordinate(mesh)
u_initial = interpolate(as_vector((exact_u(x), 0)), V)
q_initial = interpolate(firedrake.Constant(0), Q)
h = interpolate(h0 - δh * x / Lx, Q)
def viscosity(u, h, q):
A = A0 * firedrake.exp(-q / n)
return icepack.models.viscosity.viscosity_depth_averaged(u, h, A)
ice_shelf = icepack.models.IceShelf(viscosity=viscosity)
dirichlet_ids = [1, 3, 4]
tol = 1e-12
r = firedrake.sqrt((x / Lx - 1 / 2)**2 + (y / Ly - 1 / 2)**2)
R = 1 / 4
expr = firedrake.max_value(0, δA * (1 - (r / R)**2))
q_true = firedrake.interpolate(-n * firedrake.ln(1 + expr / A0), Q)
u_true = ice_shelf.diagnostic_solve(h=h,
q=q_true,
u0=u_initial,
dirichlet_ids=dirichlet_ids,
tol=tol)
# Make the noise equal to 1% of the signal
area = firedrake.assemble(firedrake.Constant(1) * dx(mesh))
σ = 0.001 * icepack.norm(u_true) / np.sqrt(area)
print(σ)
u_obs = u_true.copy(deepcopy=True)
shape = u_obs.dat.data_ro.shape
u_obs.dat.data[:] += σ * random.standard_normal(shape) / np.sqrt(2)
def callback(inverse_solver):
E, R = inverse_solver.objective, inverse_solver.regularization
misfit = firedrake.assemble(E) / area
regularization = firedrake.assemble(R) / area
q = inverse_solver.parameter
error = firedrake.norm(q - q_true) / np.sqrt(area)
print(misfit, regularization, error, flush=True)
L = 0.25 * Lx
regularization = L**2 / 2 * inner(grad(q_initial), grad(q_initial)) * dx
problem = icepack.inverse.InverseProblem(
model=ice_shelf,
method=icepack.models.IceShelf.diagnostic_solve,
objective=lambda u: 0.5 * ((u - u_obs) / σ)**2 * dx,
regularization=lambda q: 0.5 * L**2 * inner(grad(q), grad(q)) * dx,
state_name='u',
state=u_initial,
parameter_name='q',
parameter=q_initial,
model_args={
'h': h,
'u0': u_initial,
'tol': tol
},
dirichlet_ids=dirichlet_ids)
solver = solver_type(problem, callback)
max_iterations = 100
iterations = solver.solve(rtol=1e-2, atol=0, max_iterations=max_iterations)
print('Number of iterations: {}'.format(iterations))
assert iterations < max_iterations
q = solver.parameter
assert firedrake.norm(q - q_true) / firedrake.norm(q_initial -
q_true) < 1 / 3
def test_ice_shelf_inverse_with_noise(solver_type):
import icepack
Nx, Ny = 32, 32
Lx, Ly = 20e3, 20e3
u0 = 100
h0, δh = 500, 100
T0, δT = 254.15, 5.0
A0 = icepack.rate_factor(T0)
δA = icepack.rate_factor(T0 + δT) - A0
def exact_u(x):
ρ = ρ_I * (1 - ρ_I / ρ_W)
Z = icepack.rate_factor(T0) * (ρ * g * h0 / 4)**n
q = 1 - (1 - (δh / h0) * (x / Lx))**(n + 1)
δu = Z * q * Lx * (h0 / δh) / (n + 1)
return u0 + δu
mesh = firedrake.RectangleMesh(Nx, Ny, Lx, Ly)
degree = 2
V = firedrake.VectorFunctionSpace(mesh, "CG", degree)
Q = firedrake.FunctionSpace(mesh, "CG", degree)
x, y = firedrake.SpatialCoordinate(mesh)
u_initial = interpolate(as_vector((exact_u(x), 0)), V)
q_initial = interpolate(firedrake.Constant(0), Q)
h = interpolate(h0 - δh * x / Lx, Q)
def viscosity(**kwargs):
u = kwargs["velocity"]
h = kwargs["thickness"]
q = kwargs["log_fluidity"]
A = A0 * firedrake.exp(-q / n)
return icepack.models.viscosity.viscosity_depth_averaged(velocity=u,
thickness=h,
fluidity=A)
model = icepack.models.IceShelf(viscosity=viscosity)
dirichlet_ids = [1, 3, 4]
flow_solver = icepack.solvers.FlowSolver(model,
dirichlet_ids=dirichlet_ids)
r = firedrake.sqrt((x / Lx - 1 / 2)**2 + (y / Ly - 1 / 2)**2)
R = 1 / 4
expr = firedrake.max_value(0, δA * (1 - (r / R)**2))
q_true = firedrake.interpolate(-n * firedrake.ln(1 + expr / A0), Q)
u_true = flow_solver.diagnostic_solve(velocity=u_initial,
thickness=h,
log_fluidity=q_true)
# Make the noise equal to 1% of the signal
area = firedrake.assemble(firedrake.Constant(1) * dx(mesh))
σ = 0.001 * icepack.norm(u_true) / np.sqrt(area)
print(σ)
u_obs = u_true.copy(deepcopy=True)
shape = u_obs.dat.data_ro.shape
u_obs.dat.data[:] += σ * random.standard_normal(shape) / np.sqrt(2)
def callback(inverse_solver):
E, R = inverse_solver.objective, inverse_solver.regularization
misfit = firedrake.assemble(E) / area
regularization = firedrake.assemble(R) / area
q = inverse_solver.parameter
error = firedrake.norm(q - q_true) / np.sqrt(area)
print(misfit, regularization, error, flush=True)
L = firedrake.Constant(0.25 * Lx)
problem = icepack.inverse.InverseProblem(
model=model,
objective=lambda u: 0.5 * ((u - u_obs) / σ)**2 * dx,
regularization=lambda q: 0.5 * L**2 * inner(grad(q), grad(q)) * dx,
state_name="velocity",
state=u_initial,
parameter_name="log_fluidity",
parameter=q_initial,
solver_kwargs={"dirichlet_ids": dirichlet_ids},
diagnostic_solve_kwargs={"thickness": h},
)
solver = solver_type(problem, callback)
max_iterations = 100
iterations = solver.solve(rtol=1e-2, atol=0, max_iterations=max_iterations)
print(f"Number of iterations: {iterations}")
assert iterations < max_iterations
q = solver.parameter
assert firedrake.norm(q - q_true) / firedrake.norm(q_initial -
q_true) < 1 / 3
