def dHdc(self, I, L, c):
"""
Returns the derivative of the Hamiltonian consisting of ajoint-computed
self.Lam values w.r.t. the control variable <c>, i.e.,
dH d [ ]
-- = -- [ I + L(self.Lam) ]
dc dc [ ]
"""
s = "::: forming dHdc :::"
print_text(s, self.color())
dU = self.get_dU()
Lam = self.get_Lam()
# we need to evaluate the Hamiltonian with the values of Lam computed from
# self.dI in order to get the derivative of the Hamiltonian w.r.t. the
# control variables. Hence we need a new Lagrangian with the trial
# functions replaced with the computed Lam values.
L_lam = replace(L, {dU: Lam})
# the Hamiltonian with unknowns replaced with computed Lam :
H_lam = I + L_lam
# the derivative of the Hamiltonian w.r.t. the control variables in the
# direction of a P1 test function :
return derivative(H_lam, c, TestFunction(self.model.Q))
def solve_pressure(self, annotate=False):
"""
Solve for the Dukowicz BP pressure to model.p.
"""
model = self.model
# solve for vertical velocity :
s = "::: solving Dukowicz BP pressure :::"
print_text(s, self.color())
Q = model.Q
rhoi = model.rhoi
g = model.g
S = model.S
z = model.x[2]
p = model.p
eta_shf = self.eta_shf
eta_gnd = self.eta_gnd
w = self.w
p_shf = project(rhoi * g * (S - z) + 2 * eta_shf * w.dx(2),
annotate=annotate)
p_gnd = project(rhoi * g * (S - z) + 2 * eta_gnd * w.dx(2),
annotate=annotate)
# unify the pressure over shelves and grounded ice :
p_v = p.vector().array()
p_gnd_v = p_gnd.vector().array()
p_shf_v = p_shf.vector().array()
p_v[model.gnd_dofs] = p_gnd_v[model.gnd_dofs]
p_v[model.shf_dofs] = p_shf_v[model.shf_dofs]
model.assign_variable(p, p_v, cls=self)
def vert_integrate(self, u):
"""
Integrate <u> from the surface to the bed.
"""
s = '::: vertically integrating function :::'
print_text(s, self.D1Model_color)
ff = self.ff
Q = self.Q
phi = TestFunction(Q)
v = TrialFunction(Q)
# surface Dirichlet boundary :
def surface(x, on_boundary):
return on_boundary and x[0] == self.S_bc
# integral is zero on surface
bcs = DirichletBC(Q, 0.0, surface)
a = v.dx(0) * phi * dx
L = u * phi * dx
v = Function(Q)
solve(a == L, v, bcs, annotate=False)
print_min_max(v, 'vertically integrated function')
#print_min_max(v, 'vertically integrated %s' % u.name())
return v
def solve(self):
"""
"""
s = "::: solving 'FS_Balance' :::"
print_text(s, self.color())
model = self.model
# solve the linear system :
solve(self.M, model.tau_ii.vector(), assemble(self.tau_ii))
print_min_max(model.tau_ii, 'tau_ii')
solve(self.M, model.tau_ij.vector(), assemble(self.tau_ij))
print_min_max(model.tau_ij, 'tau_ij')
solve(self.M, model.tau_ik.vector(), assemble(self.tau_ik))
print_min_max(model.tau_ik, 'tau_ik')
solve(self.M, model.tau_ji.vector(), assemble(self.tau_ji))
print_min_max(model.tau_ji, 'tau_ji')
solve(self.M, model.tau_jj.vector(), assemble(self.tau_jj))
print_min_max(model.tau_jj, 'tau_jj')
solve(self.M, model.tau_jk.vector(), assemble(self.tau_jk))
print_min_max(model.tau_jk, 'tau_jk')
solve(self.M, model.tau_ki.vector(), assemble(self.tau_ki))
print_min_max(model.tau_ki, 'tau_ki')
solve(self.M, model.tau_kj.vector(), assemble(self.tau_kj))
print_min_max(model.tau_kj, 'tau_kj')
solve(self.M, model.tau_kk.vector(), assemble(self.tau_kk))
print_min_max(model.tau_kk, 'tau_kk')
def solve(self, annotate=False):
"""
Perform the Newton solve of the full-Stokes equations
"""
model = self.model
params = self.solve_params
# solve nonlinear system :
rtol = params['solver']['newton_solver']['relative_tolerance']
maxit = params['solver']['newton_solver']['maximum_iterations']
alpha = params['solver']['newton_solver']['relaxation_parameter']
s = "::: solving BP horizontal velocity with %i max iterations" + \
" and step size = %.1f :::"
print_text(s % (maxit, alpha), self.color())
# zero out self.velocity for good convergence for any subsequent solves,
# e.g. model.L_curve() :
model.assign_variable(self.get_U(), DOLFIN_EPS, cls=self)
# compute solution :
solve(self.mom_F == 0, self.G, J = self.mom_Jac, bcs = self.mom_bcs,
annotate = annotate, solver_parameters = params['solver'])
U, p = self.G.split()
u, v, w = split(U)
self.assx.assign(model.u, project(u, model.Q, annotate=False),
annotate=False)
self.assy.assign(model.v, project(v, model.Q, annotate=False),
annotate=False)
self.assz.assign(model.w, project(w, model.Q, annotate=False),
annotate=False)
self.assp.assign(model.p, p, annotate=False)
print_min_max(model.U3, 'U', cls=self)
print_min_max(model.p, 'p', cls=self)
def calc_functionals(self):
"""
Used to facilitate printing the objective function in adjoint solves.
"""
s = "::: calculating functionals :::"
print_text(s, cls=self)
ftnls = []
R = assemble(self.Rp, annotate=False)
print_min_max(R, 'R', cls=self)
ftnls.append(R)
J = assemble(self.Jp, annotate=False)
print_min_max(J, 'J', cls=self)
ftnls.append(J)
if self.obj_ftn_type == 'log_L2_hybrid':
J1 = assemble(self.J1, annotate=False)
print_min_max(J1, 'J1', cls=self)
ftnls.append(J1)
J2 = assemble(self.J2, annotate=False)
print_min_max(J2, 'J2', cls=self)
ftnls.append(J2)
if self.reg_ftn_type == 'TV_Tik_hybrid':
R1 = assemble(self.R1p, annotate=False)
print_min_max(R1, 'R1', cls=self)
ftnls.append(R1)
R2 = assemble(self.R2p, annotate=False)
print_min_max(R2, 'R2', cls=self)
ftnls.append(R2)
return ftnls
def calc_T_melt(self, annotate=True):
"""
Calculates pressure-melting point in model.T_melt.
"""
s = "::: calculating pressure-melting temperature :::"
print_text(s, cls=self)
model = self.model
dx = model.dx
gamma = model.gamma
T_w = model.T_w
p = model.p
u = TrialFunction(model.Q)
phi = TestFunction(model.Q)
Tm = Function(model.Q)
l = assemble((T_w - gamma * p) * phi * dx, annotate=annotate)
a = assemble(u * phi * dx, annotate=annotate)
solve(a, Tm.vector(), l, annotate=annotate)
model.assign_variable(model.T_melt, Tm, cls=self)
Tm_v = Tm.vector().array()
theta_m = 146.3*Tm_v + 7.253/2.0*Tm_v**2
model.assign_variable(model.theta_melt, theta_m, cls=self)
def save_lat_mesh(self, h5File): """ save the lateral boundary mesh to hdf5 file <h5File>. """ s = "::: writing 'latmesh' to supplied hdf5 file :::" print_text(s, cls=self.this) h5File.write(self.latmesh, 'latmesh')
def init_rho(self, rho):
"""
"""
s = "::: initializing density :::"
print_text(s, self.D1Model_color)
self.assign_variable(self.rho, rho)
print_min_max(self.rho, 'rho')
def save_bed_mesh(self, h5File): """ save the basal boundary mesh to hdf5 file <h5File>. """ s = "::: writing 'bedmesh' to supplied hdf5 file :::" print_text(s, cls=self.this) h5File.write(self.bedmesh, 'bedmesh')
def save_srf_mesh(self, h5File): """ save the surface boundary mesh to hdf5 file <h5File>. """ s = "::: writing 'srfmesh' to supplied hdf5 file :::" print_text(s, cls=self.this) h5File.write(self.srfmesh, 'srfmesh')
def set_subdomains_3(self, ff, cf, ff_acc):
"""
Set the facet subdomains to FacetFunction <ff>, and set the cell subdomains
to CellFunction <cf>, and accumulation FacetFunction to <ff_acc>.
"""
s = "::: setting 3D subdomains :::"
print_text(s, cls=self)
self.ff = ff
self.cf = cf
self.ff_acc = ff_acc
self.ds = Measure('ds')[self.ff]
self.dx = Measure('dx')[self.cf]
self.dx_g = self.dx(0) # internal above grounded
self.dx_f = self.dx(1) # internal above floating
self.dBed_g = self.ds(3) # grounded bed
self.dBed_f = self.ds(5) # floating bed
self.dBed = self.ds(3) + self.ds(5) # bed
self.dSrf_g = self.ds(2) # surface of grounded ice
self.dSrf_f = self.ds(6) # surface of floating ice
self.dSrf = self.ds(6) + self.ds(2) # surface
self.dLat_d = self.ds(7) # lateral divide
self.dLat_to = self.ds(4) # lateral terminus overwater
self.dLat_tu = self.ds(10) # lateral terminus underwater
self.dLat_t = self.ds(4) + self.ds(10) # lateral terminus
self.dLat = self.ds(4) + self.ds(7) \
+ self.ds(10) # lateral
def generate_stokes_function_spaces(self, kind='mini'):
"""
Generates the appropriate finite-element function spaces from parameters
specified in the config file for the model.
If <kind> == 'mini', use enriched mini elements.
If <kind> == 'th', use Taylor-Hood elements.
"""
# mini elements :
if kind == 'mini':
s = "::: generating 'mini' Stokes function spaces :::"
self.Bub = FunctionSpace(self.mesh, "B", 4,
constrained_domain=self.pBC)
self.MQ = self.Q + self.Bub
M3 = MixedFunctionSpace([self.MQ]*3)
self.Q4 = MixedFunctionSpace([M3, self.Q])
self.Q5 = MixedFunctionSpace([M3, self.Q, self.Q])
# Taylor-Hood elements :
elif kind == 'th':
s = "::: generating 'Taylor-Hood' Stokes function spaces :::"
V = VectorFunctionSpace(self.mesh, "CG", 2,
constrained_domain=self.pBC)
self.Q4 = V * self.Q
self.Q5 = V * self.Q * self.Q
else:
s = ">>> METHOD generate_stokes_function_spaces <kind> FIELD <<<\n" + \
">>> MAY BE 'mini' OR 'th', NOT '%s'. <<<" % kind
print_text(s, 'red', 1)
sys.exit(1)
print_text(s, cls=self)
def solve(self, annotate=False):
"""
Perform the Newton solve of the full-Stokes equations
"""
model = self.model
params = self.solve_params
# solve nonlinear system :
rtol = params['solver']['newton_solver']['relative_tolerance']
maxit = params['solver']['newton_solver']['maximum_iterations']
alpha = params['solver']['newton_solver']['relaxation_parameter']
s = "::: solving Dukowicz-Brinkerhoff-full-Stokes equations" + \
" with %i max iterations and step size = %.1f :::"
print_text(s % (maxit, alpha), self.color())
# zero out self.velocity for good convergence for any subsequent solves,
# e.g. model.L_curve() :
model.assign_variable(self.get_U(), DOLFIN_EPS, cls=self)
# compute solution :
solve(self.mom_F == 0, self.U, J = self.mom_Jac, bcs = self.mom_bcs,
annotate = annotate, solver_parameters = params['solver'])
u, v, w, p = self.U.split()
self.assx.assign(model.u, u, annotate=False)
self.assy.assign(model.v, v, annotate=False)
self.assz.assign(model.w, w, annotate=False)
self.assp.assign(model.p, p, annotate=False)
U3 = model.U3.split(True)
print_min_max(U3[0], 'u', cls=self)
print_min_max(U3[1], 'v', cls=self)
print_min_max(U3[2], 'w', cls=self)
print_min_max(model.p, 'p', cls=self)
def init_r(self, r):
"""
"""
s = "::: initializing grain-size :::"
print_text(s, self.D1Model_color)
self.assign_variable(self.r, r)
print_min_max(self.r, 'r^2')
def save_dvd_mesh(self, h5File): """ save the divide boundary mesh to hdf5 file <h5File>. """ s = "::: writing 'dvdmesh' to supplied hdf5 file :::" print_text(s, cls=self.this) h5File.write(self.dvdmesh, 'dvdmesh')
def solve_basal_melt_rate(self):
"""
Solve for the basal melt rate stored in model.Mb.
"""
# calculate melt-rate :
s = "::: solving for basal melt-rate :::"
print_text(s, cls=self)
model = self.model
B = model.B
rhoi = model.rhoi
theta = model.theta
T = model.T
T_melt = model.T_melt
L = model.L
q_geo = model.q_geo
rho = self.rho
k = self.k
c = self.c
q_fric = self.q_fric
gradB = as_vector([B.dx(0), B.dx(1), -1])
dTdn = k/c * dot(grad(theta), gradB)
nMb = project((q_geo + q_fric - dTdn) / (L*rho), model.Q,
annotate=False)
nMb_v = nMb.vector().array()
T_melt_v = T_melt.vector().array()
T_v = T.vector().array()
nMb_v[T_v < T_melt_v] = 0.0 # if frozen, no melt
nMb_v[model.shf_dofs] = 0.0 # does apply over floating regions
model.assign_variable(model.Mb, nMb_v, cls=self)
def vert_integrate(self, u, d='up', Q='self'):
"""
Integrate <u> from the bed to the surface.
"""
s = "::: vertically integrating function :::"
print_text(s, cls=self)
if type(Q) != FunctionSpace:
Q = self.Q
ff = self.ff
phi = TestFunction(Q)
v = TrialFunction(Q)
bcs = []
# integral is zero on bed (ff = 3,5)
if d == 'up':
bcs.append(DirichletBC(Q, 0.0, ff, self.GAMMA_B_GND)) # grounded
bcs.append(DirichletBC(Q, 0.0, ff, self.GAMMA_B_FLT)) # shelves
a = v.dx(2) * phi * dx
# integral is zero on surface (ff = 2,6)
elif d == 'down':
bcs.append(DirichletBC(Q, 0.0, ff, self.GAMMA_S_GND)) # grounded
bcs.append(DirichletBC(Q, 0.0, ff, self.GAMMA_S_FLT)) # shelves
bcs.append(DirichletBC(Q, 0.0, ff, self.GAMMA_U_GND)) # grounded
bcs.append(DirichletBC(Q, 0.0, ff, self.GAMMA_U_FLT)) # shelves
a = -v.dx(2) * phi * dx
L = u * phi * dx
name = 'value integrated %s' % d
v = Function(Q, name=name)
solve(a == L, v, bcs, annotate=False)
print_min_max(u, 'vertically integrated function', cls=self)
return v
def init_B_bc(self, B_bc):
"""
Set the scalar-value for bed.
"""
s = "::: initializng bed boundary condition :::"
print_text(s, self.D1Model_color)
self.B_bc = B_bc
print_min_max(self.B_bc, 'B_bc')
def __init__(self, mesh, out_dir='./results/', use_periodic=False):
"""
Create and instance of a 2D model.
"""
s = "::: INITIALIZING 2D MODEL :::"
print_text(s, cls=self)
Model.__init__(self, mesh, out_dir, use_periodic)
def init_S_bc(self, S_bc):
"""
Set the scalar-value for the surface.
"""
s = "::: initializng surface boundary condition :::"
print_text(s, self.D1Model_color)
self.S_bc = S_bc
print_min_max(self.S_bc, 'S_bc')
def init_U(self, U, cls=None):
"""
"""
# NOTE: this overides model.init_U
if cls is None:
cls = self.this
s = "::: initializing velocity :::"
print_text(s, cls=cls)
self.assign_variable(self.U3, U, cls=cls)
def Hamiltonian(self, I):
"""
Returns the Hamiltonian of the momentum equations with objective function
<I>.
"""
s = "::: forming Hamiltonian :::"
print_text(s, self.color())
# the Hamiltonian :
return I + self.Lagrangian()
def init_H_bounds(self, H_min, H_max, cls=None):
"""
"""
if cls is None:
cls = self
s = "::: initializing bounds on thickness :::"
print_text(s, cls=cls)
self.assign_variable(self.H_min, H_min, cls=cls)
self.assign_variable(self.H_max, H_max, cls=cls)
def deviatoric_stress_tensor(self):
"""
return the Cauchy stress tensor.
"""
s = "::: forming the deviatoric part of the Cauchy stress tensor :::"
print_text(s, self.color)
epi = self.strain_rate_tensor()
tau = 2 * self.eta * epi
return tau
def init_adot(self, adot):
"""
"""
s = "::: initializing accumulation :::"
print_text(s, self.D1Model_color)
self.assign_variable(self.adot, adot)
self.assign_variable(self.bdot, self.rhoi(0) * adot / self.spy(0))
print_min_max(self.adot, 'adot')
print_min_max(self.bdot, 'bdot')
def init_H_H0(self, H, cls=None):
"""
"""
if cls is None:
cls = self
s = "::: initializing thickness :::"
print_text(s, cls=cls)
self.assign_variable(self.H, H, cls=cls)
self.assign_variable(self.H0, H, cls=cls)
def __init__(self, mesh, out_dir='./results/', use_periodic=False):
"""
Create and instance of a 1D model.
"""
self.D1Model_color = '150'
s = "::: INITIALIZING 1D MODEL :::"
print_text(s, self.D1Model_color)
Model.__init__(self, mesh, out_dir, use_periodic)
def calc_bulk_density(self): """ Calculate the bulk density stored in model.rho_b. """ # calculate bulk density : s = "::: calculating bulk density :::" print_text(s, cls=self) model = self.model rho_b = project(self.rho, annotate=False) model.assign_variable(model.rhob, rho_b, cls=self)
def __init__(self, model):
"""
"""
s = "::: INITIALIZING FIRN MASS-BALANCE PHYSICS :::"
print_text(s, self.color())
if type(model) != D1Model:
s = ">>> FirnMass REQUIRES A 'D1Model' INSTANCE, NOT %s <<<"
print_text(s % type(model), 'red', 1)
sys.exit(1)