def set_solver_alpha_snes2(self):
V = self.alpha.function_space()
denergy = derivative(self.energy, self.alpha, TestFunction(V))
ddenergy = derivative(denergy, self.alpha, TrialFunction(V))
self.lb = self.alpha_init # interpolate(Constant("0."), V)
ub = interpolate(Constant("1."), V)
self.problem_alpha = NonlinearVariationalProblem(denergy,
self.alpha,
self.bcs_alpha,
J=ddenergy)
self.problem_alpha.set_bounds(self.lb, ub)
self.problem_alpha.lb = self.lb
# set up the solver
solver = NonlinearVariationalSolver(self.problem_alpha)
snes_solver_parameters_bounds = {
"nonlinear_solver": "snes",
"snes_solver": {
"linear_solver": "mumps",
"maximum_iterations": 300,
"report": True,
"line_search": "basic",
"method": "vinewtonrsls",
"absolute_tolerance": 1e-5,
"relative_tolerance": 1e-5,
"solution_tolerance": 1e-5
}
}
solver.parameters.update(snes_solver_parameters_bounds)
#solver.solve()
self.solver = solver
def __init__(self, energy, u, bcs, nullspace=None):
NonlinearProblem.__init__(self)
self.z = u
self.bcs = bcs
self.nullspace = nullspace
self.residual = derivative(energy, self.z,
TestFunction(self.z.function_space()))
self.jacobian = derivative(self.residual, self.z,
TrialFunction(self.z.function_space()))
def tangent(problem, solution, v, w, oldmu, newmu, k, params, task, vi, hint=None):
if k == 1:
return (hint, 0, 0)
info_green("Attempting tangent linearisation")
coldmu = backend.Constant(float(oldmu))
chgmu = backend.Constant(float(newmu)-float(oldmu))
rho = solution.split(deepcopy=True)[0]
Z = solution.function_space()
(lb, ub) = problem.bounds(Z, newmu, params)
vi_du = None
# FIXME: cache the symbolic calculation once, it can be expensive sometimes
du = backend.Function(Z)
#du = solution.copy(deepcopy=True)
F = problem.residual(solution, v, lb, ub, coldmu, params)
G = derivative(F, solution, du) + derivative(F, coldmu, chgmu)
J = problem.jacobian(G, du, params, v, w)
# FIXME: figure out if the boundary conditions depend on
# the parameters, and set the boundary conditions on the update
dubcs = problem.boundary_conditions(Z, newmu)
[dubc.homogenize() for dubc in dubcs]
# dm = problem._dm
# FIXME: make this optional
# teamno = 0
# FIXME: there's probably a more elegant way to do this.
# Or should we use one semismooth Newton step? After all
# we already have a Newton linearisation.
newproblem = BensonMunson(problem)
sp = problem.solver_parameters(float(oldmu), 0, task, params)
(success, iters, liters) = newton(G, J, du, dubcs,
newproblem.solver,
params,
sp,
None, None, problem._dm, vi_du)
hint = [None, float(oldmu)]
if not success:
#do nothing
return (hint, iters, liters)
else:
solution.assign(solution + du)
infeasibility = assemble(problem.infeasibility(solution, lb, ub, newmu, params))
info_green("Tangent linearisation success, infeasibility of guess %.5e" % (infeasibility))
return (hint, iters, liters)
def problem():
mesh = dolfin.UnitCubeMesh(MPI.comm_world, 10, 10, 10)
cell = mesh.ufl_cell()
vec_element = dolfin.VectorElement("Lagrange", cell, 1)
# scl_element = dolfin.FiniteElement("Lagrange", cell, 1)
Q = dolfin.FunctionSpace(mesh, vec_element)
# Qs = dolfin.FunctionSpace(mesh, scl_element)
# Coefficients
v = dolfin.function.argument.TestFunction(Q) # Test function
du = dolfin.function.argument.TrialFunction(Q) # Incremental displacement
u = dolfin.Function(Q) # Displacement from previous iteration
B = dolfin.Constant((0.0, -0.5, 0.0), cell) # Body force per unit volume
T = dolfin.Constant((0.1, 0.0, 0.0),
cell) # Traction force on the boundary
# B, T = dolfin.Function(Q), dolfin.Function(Q)
# Kinematics
d = u.geometric_dimension()
F = ufl.Identity(d) + grad(u) # Deformation gradient
C = F.T * F # Right Cauchy-Green tensor
# Invariants of deformation tensors
Ic = tr(C)
J = det(F)
# Elasticity parameters
E, nu = 10.0, 0.3
mu = dolfin.Constant(E / (2 * (1 + nu)), cell)
lmbda = dolfin.Constant(E * nu / ((1 + nu) * (1 - 2 * nu)), cell)
# mu, lmbda = dolfin.Function(Qs), dolfin.Function(Qs)
# Stored strain energy density (compressible neo-Hookean model)
psi = (mu / 2) * (Ic - 3) - mu * ln(J) + (lmbda / 2) * (ln(J))**2
# Total potential energy
Pi = psi * dx - dot(B, u) * dx - dot(T, u) * ds
# Compute first variation of Pi (directional derivative about u in the direction of v)
F = ufl.derivative(Pi, u, v)
# Compute Jacobian of F
J = ufl.derivative(F, u, du)
return J, F
def hyperelasticity(domain, q, p, nf=0):
# Based on https://github.com/firedrakeproject/firedrake-bench/blob/experiments/forms/firedrake_forms.py
V = ufl.FunctionSpace(domain, ufl.VectorElement('P', domain.ufl_cell(), q))
P = ufl.FunctionSpace(domain, ufl.VectorElement('P', domain.ufl_cell(), p))
v = ufl.TestFunction(V)
du = ufl.TrialFunction(V) # Incremental displacement
u = ufl.Coefficient(V) # Displacement from previous iteration
B = ufl.Coefficient(V) # Body force per unit mass
# Kinematics
I = ufl.Identity(domain.ufl_cell().topological_dimension())
F = I + ufl.grad(u) # Deformation gradient
C = F.T*F # Right Cauchy-Green tensor
E = (C - I)/2 # Euler-Lagrange strain tensor
E = ufl.variable(E)
# Material constants
mu = ufl.Constant(domain) # Lame's constants
lmbda = ufl.Constant(domain)
# Strain energy function (material model)
psi = lmbda/2*(ufl.tr(E)**2) + mu*ufl.tr(E*E)
S = ufl.diff(psi, E) # Second Piola-Kirchhoff stress tensor
PK = F*S # First Piola-Kirchoff stress tensor
# Variational problem
it = ufl.inner(PK, ufl.grad(v)) - ufl.inner(B, v)
f = [ufl.Coefficient(P) for _ in range(nf)]
return ufl.derivative(reduce(ufl.inner,
list(map(ufl.div, f)) + [it])*ufl.dx, u, du)
def derivative(form, u, du=None, coefficient_derivatives=None):
if du is None:
# Get existing arguments from form and position the new one with the next argument number
from ufl.algorithms import extract_arguments
form_arguments = extract_arguments(form)
number = max([-1] + [arg.number() for arg in form_arguments]) + 1
if any(arg.part() is not None for arg in form_arguments):
cpp.dolfin_error(
"formmanipulation.py", "compute derivative of form",
"Cannot automatically create third argument using parts, please supply one"
)
part = None
if isinstance(u, Function):
V = u.function_space()
du = Argument(V, number, part)
elif isinstance(u, (list, tuple)) and all(
isinstance(w, Function) for w in u):
V = MixedFunctionSpace([w.function_space() for w in u])
du = ufl.split(Argument(V, number, part))
else:
cpp.dolfin_error(
"formmanipulation.py", "compute derivative of form",
"Cannot automatically create third argument, please supply one"
)
return ufl.derivative(form, u, du, coefficient_derivatives)
def test_assemble_derivatives():
"""This test checks the original_coefficient_positions, which may change
under differentiation (some coefficients and constants are
eliminated)"""
mesh = UnitSquareMesh(MPI.COMM_WORLD, 12, 12)
Q = dolfinx.FunctionSpace(mesh, ("Lagrange", 1))
u = dolfinx.Function(Q)
v = ufl.TestFunction(Q)
du = ufl.TrialFunction(Q)
b = dolfinx.Function(Q)
c1 = fem.Constant(mesh, [[1.0, 0.0], [3.0, 4.0]])
c2 = fem.Constant(mesh, 2.0)
with b.vector.localForm() as b_local:
b_local.set(2.0)
# derivative eliminates 'u' and 'c1'
L = ufl.inner(c1, c1) * v * dx + c2 * b * inner(u, v) * dx
a = derivative(L, u, du)
A1 = dolfinx.fem.assemble_matrix(a)
A1.assemble()
a = c2 * b * inner(du, v) * dx
A2 = dolfinx.fem.assemble_matrix(a)
A2.assemble()
assert (A1 - A2).norm() == pytest.approx(0.0, rel=1e-12, abs=1e-12)
def split_arity(form, x, argument):
if x not in form.coefficients():
# No dependence on x
return ufl.classes.Form([]), form
form_derivative = ufl.derivative(form, x, argument=argument)
form_derivative = ufl.algorithms.expand_derivatives(form_derivative)
if x in form_derivative.coefficients():
# Non-linear
return ufl.classes.Form([]), form
arity = len(form.arguments())
try:
eq_form = ufl.algorithms.expand_derivatives(
ufl.replace(form, {x: argument}))
A = ufl.algorithms.formtransformations.compute_form_with_arity(
eq_form, arity + 1)
b = ufl.algorithms.formtransformations.compute_form_with_arity(
eq_form, arity)
except ufl.UFLException:
# UFL error encountered
return ufl.classes.Form([]), form
if not is_cached(A):
# Non-cached higher arity form
return ufl.classes.Form([]), form
# Success
return A, b
def evaluate_adj_component(self,
inputs,
adj_inputs,
block_variable,
idx,
prepared=None):
if not self.lincom:
if isinstance(block_variable.output,
(AdjFloat, self.backend.Constant)):
return adj_inputs[0].sum()
else:
adj_output = self.backend.Function(
block_variable.output.function_space())
adj_output.assign(prepared)
return adj_output.vector()
else:
# Linear combination
expr, adj_input_func = prepared
adj_output = self.backend.Function(
block_variable.output.function_space())
diff_expr = ufl.algorithms.expand_derivatives(
ufl.derivative(expr, block_variable.saved_output,
adj_input_func))
adj_output.assign(diff_expr)
return adj_output.vector()
def derivative(form, u, du=None, coefficient_derivatives=None):
"""Compute the derivative of a form.
Given a form, this computes its linearization with respect to the
provided :class:`.Function`. The resulting form has one
additional :class:`Argument` in the same finite element space as
the Function.
:arg form: a :class:`~ufl.classes.Form` to compute the derivative of.
:arg u: a :class:`.Function` to compute the derivative with
respect to.
:arg du: an optional :class:`Argument` to use as the replacement
in the new form (constructed automatically if not provided).
:arg coefficient_derivatives: an optional :class:`dict` to
provide the derivative of a coefficient function.
:raises ValueError: If any of the coefficients in ``form`` were
obtained from ``u.split()``. UFL doesn't notice that these
are related to ``u`` and so therefore the derivative is
wrong (instead one should have written ``split(u)``).
See also :func:`ufl.derivative`.
"""
#EVENTUALLY ADD THIS CHECKING BACK IN!
# if set(extract_coefficients(form)) & set(u.split()):
# raise ValueError("Taking derivative of form wrt u, but form contains coefficients from u.split()."
# "\nYou probably meant to write split(u) when defining your form.")
if du is None:
if isinstance(u, Function):
V = u.function_space()
args = form.arguments()
number = max(a.number() for a in args) if args else -1
du = Argument(V, number + 1)
else:
raise RuntimeError("Can't compute derivative for form")
return ufl.derivative(form, u, du, coefficient_derivatives)
def derivative(form, u, du=None, coefficient_derivatives=None):
if du is None:
# Get existing arguments from form and position the new one
# with the next argument number
form_arguments = form.arguments()
number = max([-1] + [arg.number() for arg in form_arguments]) + 1
if any(arg.part() is not None for arg in form_arguments):
cpp.dolfin_error("formmanipulation.py",
"compute derivative of form",
"Cannot automatically create new Argument using "
"parts, please supply one")
part = None
if isinstance(u, Function):
V = u.function_space()
du = Argument(V, number, part)
elif isinstance(u, (list, tuple)) and all(isinstance(w, Function) for w in u):
cpp.dolfin_error("formmanipulation.py",
"take derivative of form w.r.t. a tuple of Coefficients",
"Take derivative w.r.t. a single Coefficient on "
"a mixed space instead.")
else:
cpp.dolfin_error("formmanipulation.py",
"compute derivative of form w.r.t. '%s'" % u,
"Supply Function as a Coefficient")
return ufl.derivative(form, u, du, coefficient_derivatives)
def derivative(form, u, du=None, coefficient_derivatives=None):
if du is None:
# Get existing arguments from form and position the new one
# with the next argument number
form_arguments = form.arguments()
number = max([-1] + [arg.number() for arg in form_arguments]) + 1
if any(arg.part() is not None for arg in form_arguments):
raise RuntimeError(
"Cannot automatically create new Argument using parts, please supply one"
)
part = None
if isinstance(u, Function):
V = u.function_space()
du = Argument(V, number, part)
elif isinstance(u, (list, tuple)) and all(
isinstance(w, Function) for w in u):
raise RuntimeError(
"Take derivative w.r.t. a single Coefficient on a mixed space instead."
)
else:
raise RuntimeError(
"Computing derivative of form w.r.t. '{}'. Supply Function as a Coefficient"
.format(u))
return ufl.derivative(form, u, du, coefficient_derivatives)
def derivative(form, u, du=None, coefficient_derivatives=None):
if du is None:
# Get existing arguments from form and position the new one with the next argument number
from ufl.algorithms import extract_arguments
form_arguments = extract_arguments(form)
number = max([-1] + [arg.number() for arg in form_arguments]) + 1
if any(arg.part() is not None for arg in form_arguments):
cpp.dolfin_error("formmanipulation.py",
"compute derivative of form",
"Cannot automatically create third argument using parts, please supply one")
part = None
if isinstance(u, Function):
V = u.function_space()
du = Argument(V, number, part)
elif isinstance(u, (list,tuple)) and all(isinstance(w, Function) for w in u):
V = MixedFunctionSpace([w.function_space() for w in u])
du = ufl.split(Argument(V, number, part))
else:
cpp.dolfin_error("formmanipulation.py",
"compute derivative of form",
"Cannot automatically create third argument, please supply one")
return ufl.derivative(form, u, du, coefficient_derivatives)
def derivative(form, u, du=None, coefficient_derivatives=None):
"""Compute the derivative of a form.
Given a form, this computes its linearization with respect to the
provided :class:`.Function`. The resulting form has one
additional :class:`Argument` in the same finite element space as
the Function.
:arg form: a :class:`~ufl.classes.Form` to compute the derivative of.
:arg u: a :class:`.Function` to compute the derivative with
respect to.
:arg du: an optional :class:`Argument` to use as the replacement
in the new form (constructed automatically if not provided).
:arg coefficient_derivatives: an optional :class:`dict` to
provide the derivative of a coefficient function.
See also :func:`ufl.derivative`.
"""
if du is None:
if isinstance(u, function.Function):
V = u.function_space()
args = form.arguments()
number = max(a.number() for a in args) if args else -1
du = Argument(V, number + 1)
else:
raise RuntimeError("Can't compute derivative for form")
return ufl.derivative(form, u, du, coefficient_derivatives)
def derivative(form, u, du=None, coefficient_derivatives=None):
if du is None:
# Get existing arguments from form and position the new one
# with the next argument number
form_arguments = form.arguments()
number = max([-1] + [arg.number() for arg in form_arguments]) + 1
# NOTE : Mixed-domains problems need to have arg.part() != None
# if any(arg.part() is not None for arg in form_arguments):
# raise RuntimeError("Compute derivative of form, cannot automatically create new Argument using parts, please supply one")
part = None
if isinstance(u, Function):
# u.part() is None except with mixed-domains
part = u.part()
V = u.function_space()
du = Argument(V, number, part)
elif isinstance(u, SpatialCoordinate):
mesh = u.ufl_domain().ufl_cargo()
element = u.ufl_domain().ufl_coordinate_element()
V = FunctionSpace(mesh, element)
du = Argument(V, number, part)
elif isinstance(u, (list, tuple)) and all(
isinstance(w, Function) for w in u):
raise RuntimeError(
"Taking derivative of form w.r.t. a tuple of Coefficients. Take derivative w.r.t. a single Coefficient on a mixed space instead."
)
else:
raise RuntimeError(
"Computing derivative of form w.r.t. '{}'. Supply Function as a Coefficient"
.format(u))
return ufl.derivative(form, u, du, coefficient_derivatives)
def test_assemble_derivatives():
"""This test checks the original_coefficient_positions, which may change
under differentiation (some coefficients and constants are
eliminated)"""
mesh = create_unit_square(MPI.COMM_WORLD, 12, 12)
Q = FunctionSpace(mesh, ("Lagrange", 1))
u = Function(Q)
v = ufl.TestFunction(Q)
du = ufl.TrialFunction(Q)
b = Function(Q)
c1 = Constant(mesh, np.array([[1.0, 0.0], [3.0, 4.0]], PETSc.ScalarType))
c2 = Constant(mesh, PETSc.ScalarType(2.0))
b.x.array[:] = 2.0
# derivative eliminates 'u' and 'c1'
L = ufl.inner(c1, c1) * v * dx + c2 * b * inner(u, v) * dx
a = form(derivative(L, u, du))
A1 = assemble_matrix(a)
A1.assemble()
a = form(c2 * b * inner(du, v) * dx)
A2 = assemble_matrix(a)
A2.assemble()
assert (A1 - A2).norm() == pytest.approx(0.0, rel=1e-12, abs=1e-12)
def derivative(form: ufl.Form, u, du,
coefficient_derivatives=None) -> ufl.Form:
"""Compute derivative of from about u (coefficient) in the direction
of du (Argument)
"""
return ufl.derivative(form, u, du, coefficient_derivatives)
def derivative(form, u, du=None, coefficient_derivatives=None):
if du is None:
# Get existing arguments from form and position the new one
# with the next argument number
form_arguments = form.arguments()
number = max([-1] + [arg.number() for arg in form_arguments]) + 1
if any(arg.part() is not None for arg in form_arguments):
raise RuntimeError("Compute derivative of form, cannot automatically create new Argument using parts, please supply one")
part = None
if isinstance(u, Function):
V = u.function_space()
du = Argument(V, number, part)
elif isinstance(u, SpatialCoordinate):
mesh = u.ufl_domain().ufl_cargo()
element = u.ufl_domain().ufl_coordinate_element()
V = FunctionSpace(mesh, element)
du = Argument(V, number, part)
elif isinstance(u, (list, tuple)) and all(isinstance(w, Function) for w in u):
raise RuntimeError("Taking derivative of form w.r.t. a tuple of Coefficients. Take derivative w.r.t. a single Coefficient on a mixed space instead.")
else:
raise RuntimeError("Computing derivative of form w.r.t. '{}'. Supply Function as a Coefficient".format(u))
return ufl.derivative(form, u, du, coefficient_derivatives)
def derivative(form, u, du=None, coefficient_derivatives=None):
"""Compute the derivative of a form.
Given a form, this computes its linearization with respect to the
provided :class:`.Function`. The resulting form has one
additional :class:`Argument` in the same finite element space as
the Function.
:arg form: a :class:`ufl.Form` to compute the derivative of.
:arg u: a :class:`.Function` to compute the derivative with
respect to.
:arg du: an optional :class:`Argument` to use as the replacement
in the new form (constructed automatically if not provided).
:arg coefficient_derivatives: an optional :class:`dict` to
provide the derivative of a coefficient function.
See also :func:`ufl.derivative`.
"""
if du is None:
if isinstance(u, function.Function):
V = u.function_space()
args = form.arguments()
number = max(a.number() for a in args) if args else -1
du = Argument(V, number + 1)
else:
raise RuntimeError("Can't compute derivative for form")
return ufl.derivative(form, u, du, coefficient_derivatives)
def differentiate_expr(expr, u, expand = True):
"""
Wrapper for the UFL derivative function. This chooses an argument equal to
Constant(1.0). Form s should be differentiated using the derivative function.
"""
if not isinstance(expr, ufl.expr.Expr):
raise InvalidArgumentException("expr must be an Expr")
if isinstance(u, ufl.indexed.Indexed):
op = u.operands()
assert(len(op) == 2)
if not isinstance(op[0], (dolfin.Constant, dolfin.Function)):
raise InvalidArgumentException("Invalid Indexed")
elif not isinstance(u, (dolfin.Constant, dolfin.Function)):
raise InvalidArgumentException("u must be an Indexed, Constant, or Function")
if expr is u:
der = ufl.constantvalue.IntValue(1)
else:
unity = dolfin.Constant(1.0)
der = dolfin.replace(ufl.derivative(expr, u, argument = unity), {unity:ufl.constantvalue.IntValue(1)})
if expand:
# Based on code from expand_derivatives1 in UFL file ad.py, (see e.g. bzr
# 1.1.x branch revision 1484)
cell = der.cell()
if cell is None:
dim = 0
else:
dim = der.cell().geometric_dimension()
der = ufl.algorithms.expand_derivatives(der, dim = dim)
return der
def differentiate_expr(expr, u, expand = True):
"""
Wrapper for the UFL derivative function. This chooses an argument equal to
Constant(1.0). Form s should be differentiated using the derivative function.
"""
if not isinstance(expr, ufl.core.expr.Expr):
raise InvalidArgumentException("expr must be an Expr")
if isinstance(u, ufl.indexed.Indexed):
op = u.operands()
assert(len(op) == 2)
if not isinstance(op[0], (dolfin.Constant, dolfin.Function)):
raise InvalidArgumentException("Invalid Indexed")
elif not isinstance(u, (dolfin.Constant, dolfin.Function)):
raise InvalidArgumentException("u must be an Indexed, Constant, or Function")
if expr is u:
der = ufl.constantvalue.IntValue(1)
else:
unity = dolfin.Constant(1.0)
der = dolfin.replace(ufl.derivative(expr, u, argument = unity), {unity:ufl.constantvalue.IntValue(1)})
if expand:
# Based on code from expand_derivatives1 in UFL file ad.py, (see e.g. bzr
# 1.1.x branch revision 1484)
cell = der.cell()
if cell is None:
dim = 0
else:
dim = der.cell().geometric_dimension()
der = ufl.algorithms.expand_derivatives(der, dim = dim)
return der
def test_pack_coefficients():
"""Test packing of form coefficients ahead of main assembly call"""
mesh = create_unit_square(MPI.COMM_WORLD, 12, 15)
V = FunctionSpace(mesh, ("Lagrange", 1))
# Non-blocked
u = Function(V)
v = ufl.TestFunction(V)
c = Constant(mesh, PETSc.ScalarType(12.0))
F = ufl.inner(c, v) * dx - c * ufl.sqrt(u * u) * ufl.inner(u, v) * dx
u.x.array[:] = 10.0
_F = form(F)
# -- Test vector
b0 = assemble_vector(_F)
b0.assemble()
constants = pack_constants(_F)
coeffs = pack_coefficients(_F)
with b0.localForm() as _b0:
for c in [(None, None), (None, coeffs), (constants, None), (constants, coeffs)]:
b = assemble_vector(_F, coeffs=c)
b.assemble()
with b.localForm() as _b:
assert (_b0.array_r == _b.array_r).all()
# Change coefficients
constants *= 5.0
for coeff in coeffs.values():
coeff *= 5.0
with b0.localForm() as _b0:
for c in [(None, coeffs), (constants, None), (constants, coeffs)]:
b = assemble_vector(_F, coeffs=c)
b.assemble()
with b.localForm() as _b:
assert (_b0 - _b).norm() > 1.0e-5
# -- Test matrix
du = ufl.TrialFunction(V)
J = ufl.derivative(F, u, du)
J = form(J)
A0 = assemble_matrix(J)
A0.assemble()
constants = pack_constants(J)
coeffs = pack_coefficients(J)
for c in [(None, None), (None, coeffs), (constants, None), (constants, coeffs)]:
A = assemble_matrix(J, coeffs=c)
A.assemble()
assert pytest.approx((A - A0).norm(), 1.0e-12) == 0.0
# Change coefficients
constants *= 5.0
for coeff in coeffs.values():
coeff *= 5.0
for c in [(None, coeffs), (constants, None), (constants, coeffs)]:
A = assemble_matrix(J, coeffs=c)
A.assemble()
assert (A - A0).norm() > 1.0e-5
def hyperelasticity_action_forms(mesh, vec_el):
cell = mesh.ufl_cell()
Q = dolfin.FunctionSpace(mesh, vec_el)
# Coefficients
v = dolfin.function.argument.TestFunction(Q) # Test function
du = dolfin.function.argument.TrialFunction(Q) # Incremental displacement
u = dolfin.Function(Q) # Displacement from previous iteration
u.vector().set(0.5)
B = dolfin.Constant((0.0, -0.5, 0.0), cell) # Body force per unit volume
T = dolfin.Constant((0.1, 0.0, 0.0), cell) # Traction force on the boundary
# Kinematics
d = u.geometric_dimension()
F = ufl.Identity(d) + grad(u) # Deformation gradient
C = F.T * F # Right Cauchy-Green tensor
# Invariants of deformation tensors
Ic = tr(C)
J = det(F)
# Elasticity parameters
E, nu = 10.0, 0.3
mu = dolfin.Constant(E / (2 * (1 + nu)), cell)
lmbda = dolfin.Constant(E * nu / ((1 + nu) * (1 - 2 * nu)), cell)
# Stored strain energy density (compressible neo-Hookean model)
psi = (mu / 2) * (Ic - 3) - mu * ln(J) + (lmbda / 2) * (ln(J)) ** 2
# Total potential energy
Pi = psi * dx - dot(B, u) * dx - dot(T, u) * ds
# Compute first variation of Pi (directional derivative about u in the direction of v)
F = ufl.derivative(Pi, u, v)
# Compute Jacobian of F
J = ufl.derivative(F, u, du)
w = dolfin.Function(Q)
w.vector().set(1.2)
L = ufl.action(J, w)
return None, L, None
def __init__(self, F, u, bc):
V = u.function_space
du = TrialFunction(V)
self.L = form(F)
self.a = form(derivative(F, u, du))
self.bc = bc
self._F, self._J = None, None
self.u = u
def derivative(form, x, argument=None):
if argument is None:
space = derivative_space(x)
rank = len(form.arguments())
Argument = {0: TestFunction, 1: TrialFunction}[rank]
argument = Argument(space)
return ufl.derivative(form, x, argument=argument)
def variation(self, v):
""" Directional derivative in direction v """
# return ufl.algorithms.expand_derivatives(ufl.derivative(self.expression, self.variable, v))
deriv = sum([
ufl.derivative(self.expression, var, v_)
for (var, v_) in zip(split(self.variable), split(v))
])
return ufl.algorithms.expand_derivatives(deriv)
def __init__(self, F, u, bc):
super().__init__()
V = u.function_space
du = TrialFunction(V)
self.L = F
self.a = derivative(F, u, du)
self.bc = bc
self._F, self._J = None, None
def derivative(e, u, du, u0=None):
"""Generate the functional derivative of UFL expression (or list of expressions) for the
given function(s) u in the direction of function(s) du and at u0.
Example (first variation): δe = dolfiny.expression.derivative(e, m, δm)
Parameters
----------
e: UFL Expr or list of UFL expressions/forms
u: UFL Function or list of UFL functions
du: UFL Function or list of UFL functions
u0: UFL Function or list of UFL functions, defaults to u
Returns
-------
Expression (or list of expressions) with functional derivative.
"""
if u0 is None:
u0 = u
e0 = evaluate(e, u, u0)
if isinstance(e0, list):
de0 = []
for e0_ in e0:
if isinstance(u0, list) and isinstance(du, list):
de0_ = sum(
ufl.derivative(e0_, v0, dv) for v0, dv in zip(u0, du))
else:
de0_ = ufl.derivative(e0_, u0, du)
de0_ = ufl.algorithms.apply_algebra_lowering.apply_algebra_lowering(
de0_)
de0_ = ufl.algorithms.apply_derivatives.apply_derivatives(de0_)
de0.append(de0_)
else:
if isinstance(u0, list) and isinstance(du, list):
de0 = sum(ufl.derivative(e0, v0, dv) for v0, dv in zip(u0, du))
else:
de0 = ufl.derivative(e0, u0, du)
de0 = ufl.algorithms.apply_algebra_lowering.apply_algebra_lowering(de0)
de0 = ufl.algorithms.apply_derivatives.apply_derivatives(de0)
return de0
def __init__(self, energy, alpha, bcs, lb=None, ub=None):
OptimisationProblem.__init__(self)
self.energy = energy
self.alpha = alpha
self.V = self.alpha.function_space()
self.denergy = derivative(self.energy, self.alpha,
TestFunction(self.V))
self.ddenergy = derivative(self.denergy, self.alpha,
TrialFunction(self.V))
if lb == None:
lb = interpolate(Constant("0."), self.V)
if ub == None:
ub = interpolate(Constant("2."), self.V)
self.lb = lb
self.ub = ub
self.bcs = bcs
self.update_lb()
def __init__(self, F, u, bc):
V = u.function_space
du = TrialFunction(V)
self.L = F
self.a = derivative(F, u, du)
self.a_comp = dolfinx.fem.Form(self.a)
self.bc = bc
self._F, self._J = None, None
self.u = u
def evaluate_adj_component(self,
inputs,
adj_inputs,
block_variable,
idx,
prepared=None):
if self.expr is None:
if isinstance(block_variable.output, AdjFloat):
return adj_inputs[0].sum()
elif isinstance(block_variable.output, self.backend.Constant):
R = block_variable.output._ad_function_space(
prepared.function_space().mesh())
return self._adj_assign_constant(prepared, R)
else:
adj_output = self.backend.Function(
block_variable.output.function_space())
adj_output.assign(prepared)
return adj_output.vector()
else:
# Linear combination
expr, adj_input_func = prepared
adj_output = self.backend.Function(adj_input_func.function_space())
if block_variable.saved_output.ufl_shape == adj_input_func.ufl_shape:
diff_expr = ufl.algorithms.expand_derivatives(
ufl.derivative(expr, block_variable.saved_output,
adj_input_func))
adj_output.assign(diff_expr)
else:
# Assume current variable is scalar constant.
# This assumption might be wrong for firedrake.
assert isinstance(block_variable.output, self.backend.Constant)
diff_expr = ufl.algorithms.expand_derivatives(
ufl.derivative(expr, block_variable.saved_output,
self.backend.Constant(1.)))
adj_output.assign(diff_expr)
return adj_output.vector().inner(adj_input_func.vector())
if isinstance(block_variable.output, self.backend.Constant):
R = block_variable.output._ad_function_space(
adj_output.function_space().mesh())
return self._adj_assign_constant(adj_output, R)
else:
return adj_output.vector()
def __init__(self, energy, alpha, bcs, lb=None, ub=None):
NonlinearProblem.__init__(self)
self.energy = energy
self.alpha = alpha
self.V = self.alpha.function_space()
self.denergy = derivative(self.energy, self.alpha,
TestFunction(self.V))
self.ddenergy = derivative(self.denergy, self.alpha,
TrialFunction(self.V))
if lb == None:
lb = interpolate(Constant("0."), self.V)
if ub == None:
ub = interpolate(Constant("1."), self.V)
self.lb = lb
self.ub = ub
self.bcs = bcs
self.b = PETScVector()
self.A = PETScMatrix()
def test_block(V1, V2, squaremesh_5, nest):
mesh = squaremesh_5
u0 = dolfinx.Function(V1, name="u0")
u1 = dolfinx.Function(V2, name="u1")
v0 = ufl.TestFunction(V1)
v1 = ufl.TestFunction(V2)
Phi = (ufl.sin(u0) - 0.5)**2 * ufl.dx(mesh) + (4.0 * u0 -
u1)**2 * ufl.dx(mesh)
F0 = ufl.derivative(Phi, u0, v0)
F1 = ufl.derivative(Phi, u1, v1)
F = [F0, F1]
u = [u0, u1]
opts = PETSc.Options("block")
opts.setValue('snes_type', 'newtontr')
opts.setValue('snes_rtol', 1.0e-08)
opts.setValue('snes_max_it', 12)
if nest:
opts.setValue('ksp_type', 'cg')
opts.setValue('pc_type', 'fieldsplit')
opts.setValue('fieldsplit_pc_type', 'lu')
opts.setValue('ksp_rtol', 1.0e-10)
else:
opts.setValue('ksp_type', 'preonly')
opts.setValue('pc_type', 'lu')
opts.setValue('pc_factor_mat_solver_type', 'mumps')
problem = dolfiny.snesblockproblem.SNESBlockProblem(F,
u,
nest=nest,
prefix="block")
sol = problem.solve()
assert problem.snes.getConvergedReason() > 0
assert np.isclose((sol[0].vector - np.arcsin(0.5)).norm(), 0.0)
assert np.isclose((sol[1].vector - 4.0 * np.arcsin(0.5)).norm(), 0.0)
def __init__(self, F, u, bc):
super().__init__()
V = u.function_space()
du = function.TrialFunction(V)
self.L = F
self.a = derivative(F, u, du)
self.a_comp = dolfin.fem.Form(self.a)
self.bc = bc
self._F, self._J = None, None
self.u = u
def derivative(form, u, du=None):
if du is None:
if isinstance(u, Function):
V = u.function_space()
du = Argument(V)
elif isinstance(u, (list,tuple)) and all(isinstance(w, Function) for w in u):
V = MixedFunctionSpace([w.function_space() for w in u])
du = ufl.split(Argument(V))
else:
cpp.dolfin_error("formmanipulation.py",
"compute derivative of form",
"Cannot automatically create third argument, please supply one")
return ufl.derivative(form, u, du)
def derivative(form, u, du=None, coefficient_derivatives=None):
if du is None:
# Get existing arguments from form and position the new one with the next argument number
form_arguments = form.arguments()
number = max([-1] + [arg.number() for arg in form_arguments]) + 1
if any(arg.part() is not None for arg in form_arguments):
cpp.dolfin_error("formmanipulation.py",
"compute derivative of form",
"Cannot automatically create new Argument using "
"parts, please supply one")
part = None
if isinstance(u, Function):
V = u.function_space()
du = Argument(V, number, part)
elif isinstance(u, (list,tuple)) and all(isinstance(w, Function) for w in u):
cpp.deprecation("derivative of form w.r.t. a tuple of Coefficients",
"1.7.0", "2.0.0",
"Take derivative w.r.t. a single Coefficient on "
"a mixed space instead.")
# Get mesh
mesh = u[0].function_space().mesh()
if not all(mesh.id() == v.function_space().mesh().id() for v in u):
cpp.dolfin_error("formmanipulation.py",
"compute derivative of form",
"Don't know how to compute derivative with "
"respect to Coefficients on different meshes yet")
# Build mixed element, space and argument
element = ufl.MixedElement([v.function_space().ufl_element() for v in u])
V = FunctionSpace(mesh, element)
du = ufl.split(Argument(V, number, part))
else:
cpp.dolfin_error("formmanipulation.py",
"compute derivative of form w.r.t. '%s'" % u,
"Supply Function as a Coefficient")
return ufl.derivative(form, u, du, coefficient_derivatives)
def derivative(form, u, du=None):
"""Compute the derivative of a form.
Given a form, this computes its linearization with respect to the
provided :class:`.Function`. The resulting form has one
additional :class:`Argument` in the same finite element space as
the Function.
:arg form: a :class:`ufl.Form` to compute the derivative of.
:arg u: a :class:`.Function` to compute the derivative with
respect to.
:arg du: an optional :class:`Argument` to use as the replacement
in the new form (constructed automatically if not provided).
See also :func:`ufl.derivative`.
"""
if du is None:
if isinstance(u, function.Function):
V = u.function_space()
du = TrialFunction(V)
else:
raise RuntimeError("Can't compute derivative for form")
return ufl.derivative(form, u, du)
def test_conditional_nan():
# Test case courtesy of Marco Morandini:
# https://github.com/firedrakeproject/tsfc/issues/183
def crossTensor(V):
return as_tensor([
[0.0, V[2], -V[1]],
[-V[2], 0.0, V[0]],
[V[1], -V[0], 0.0]])
def coefa(phi2):
return conditional(le(phi2, 1.e-6),
phi2,
sin(phi2)/phi2)
def RotDiffTensor(phi):
PhiCross = crossTensor(phi)
phi2 = dot(phi, phi)
a = coefa(phi2)
return PhiCross * a
mesh = UnitIntervalMesh(8)
V = VectorFunctionSpace(mesh, "CG", 1, dim=3)
u_phi = Function(V)
v_phi = TestFunction(V)
Gamma = RotDiffTensor(u_phi)
beta = dot(Gamma, grad(u_phi))
delta_beta = ufl.derivative(beta, u_phi, v_phi)
L = inner(delta_beta, grad(u_phi)) * dx
result = assemble(L).dat.data
# Check whether there are any NaNs in the result
assert not np.isnan(result).any()
def solve(self, problem):
self.problem = problem
doSave = problem.doSave
save_this_step = False
onlyVel = problem.saveOnlyVel
dt = self.metadata['dt']
nu = Constant(self.problem.nu)
self.tc.init_watch('init', 'Initialization', True, count_to_percent=False)
self.tc.init_watch('solve', 'Running nonlinear solver', True, count_to_percent=True)
self.tc.init_watch('next', 'Next step assignments', True, count_to_percent=True)
self.tc.init_watch('saveVel', 'Saved velocity', True)
self.tc.start('init')
mesh = self.problem.mesh
# Define function spaces (P2-P1)
self.V = VectorFunctionSpace(mesh, "Lagrange", 2) # velocity
self.Q = FunctionSpace(mesh, "Lagrange", 1) # pressure
self.W = MixedFunctionSpace([self.V, self.Q])
self.PS = FunctionSpace(mesh, "Lagrange", 2) # partial solution (must be same order as V)
self.D = FunctionSpace(mesh, "Lagrange", 1) # velocity divergence space
# to assign solution in space W.sub(0) to Function(V) we need FunctionAssigner (cannot be assigned directly)
fa = FunctionAssigner(self.V, self.W.sub(0))
velSp = Function(self.V)
problem.initialize(self.V, self.Q, self.PS, self.D)
# Define unknown and test function(s) NS
v, q = TestFunctions(self.W)
w = Function(self.W)
dw = TrialFunction(self.W)
u, p = split(w)
# Define fields
n = FacetNormal(mesh)
I = Identity(u.geometric_dimension())
theta = 0.5 # Crank-Nicholson
k = Constant(self.metadata['dt'])
# Initial conditions: u0 velocity at previous time step u1 velocity two time steps back p0 previous pressure
[u0, p0] = self.problem.get_initial_conditions([{'type': 'v', 'time': 0.0}, {'type': 'p', 'time': 0.0}])
if doSave:
problem.save_vel(False, u0, 0.0)
# boundary conditions
bcu, bcp = problem.get_boundary_conditions(self.bc == 'outflow', self.W.sub(0), self.W.sub(1))
# NT bcp is not used
# Define steady part of the equation
def T(u):
return -p * I + 2.0 * nu * sym(grad(u))
def F(u, v, q):
return (inner(T(u), grad(v)) - q * div(u)) * dx + inner(grad(u) * u, v) * dx
# Define variational forms
F_ns = (inner((u - u0), v) / k) * dx + (1.0 - theta) * F(u0, v, q) + theta * F(u, v, q)
J_ns = derivative(F_ns, w, dw)
# J_ns = derivative(F_ns, w) # did not work
# NS_problem = NonlinearVariationalProblem(F_ns, w, bcu, J_ns, form_compiler_parameters=ffc_options)
NS_problem = NonlinearVariationalProblem(F_ns, w, bcu, J_ns)
# (var. formulation, unknown, Dir. BC, jacobian, optional)
NS_solver = NonlinearVariationalSolver(NS_problem)
prm = NS_solver.parameters
prm['newton_solver']['absolute_tolerance'] = 1E-08
prm['newton_solver']['relative_tolerance'] = 1E-08
# prm['newton_solver']['maximum_iterations'] = 45
# prm['newton_solver']['relaxation_parameter'] = 1.0
prm['newton_solver']['linear_solver'] = 'mumps'
info(NS_solver.parameters, True)
self.tc.end('init')
# Time-stepping
info("Running of direct method")
ttime = self.metadata['time']
t = dt
step = 1
while t < (ttime + dt/2.0):
info("t = %f" % t)
self.problem.update_time(t, step)
if self.MPI_rank == 0:
problem.write_status_file(t)
if doSave:
save_this_step = problem.save_this_step
# Compute
begin("Solving NS ....")
try:
self.tc.start('solve')
NS_solver.solve()
self.tc.end('solve')
except RuntimeError as inst:
problem.report_fail(t)
return 1
end()
# Extract solutions:
(u, p) = w.split()
fa.assign(velSp, u)
# we are assigning twice (now and inside save_vel), but it works with one method save_vel for direct and
# projection (we could split save_vel to save one assign)
if save_this_step:
self.tc.start('saveVel')
problem.save_vel(False, velSp, t)
self.tc.end('saveVel')
if save_this_step and not onlyVel:
problem.save_div(False, u)
problem.compute_err(False, u, t)
problem.compute_div(False, u)
# foo = Function(self.Q)
# foo.assign(p)
# problem.averaging_pressure(foo)
# if save_this_step and not onlyVel:
# problem.save_pressure(False, foo)
if save_this_step and not onlyVel:
problem.save_pressure(False, p)
# compute functionals (e. g. forces)
problem.compute_functionals(u, p, t)
# Move to next time step
self.tc.start('next')
u0.assign(velSp)
t = round(t + dt, 6) # round time step to 0.000001
step += 1
self.tc.end('next')
info("Finished: direct method")
problem.report()
return 0
def derivative(form, u, du=None, coefficient_derivatives=None):
"""Compute the derivative of a form.
Given a form, this computes its linearization with respect to the
provided :class:`.Function`. The resulting form has one
additional :class:`Argument` in the same finite element space as
the Function.
:arg form: a :class:`~ufl.classes.Form` to compute the derivative of.
:arg u: a :class:`.Function` to compute the derivative with
respect to.
:arg du: an optional :class:`Argument` to use as the replacement
in the new form (constructed automatically if not provided).
:arg coefficient_derivatives: an optional :class:`dict` to
provide the derivative of a coefficient function.
:raises ValueError: If any of the coefficients in ``form`` were
obtained from ``u.split()``. UFL doesn't notice that these
are related to ``u`` and so therefore the derivative is
wrong (instead one should have written ``split(u)``).
See also :func:`ufl.derivative`.
"""
# TODO: What about Constant?
u_is_x = isinstance(u, ufl.SpatialCoordinate)
if not u_is_x and len(u.split()) > 1 and set(extract_coefficients(form)) & set(u.split()):
raise ValueError("Taking derivative of form wrt u, but form contains coefficients from u.split()."
"\nYou probably meant to write split(u) when defining your form.")
mesh = form.ufl_domain()
is_dX = u_is_x or u is mesh.coordinates
args = form.arguments()
def argument(V):
if du is None:
n = max(a.number() for a in args) if args else -1
return Argument(V, n + 1)
else:
return du
if is_dX:
coords = mesh.coordinates
u = ufl.SpatialCoordinate(mesh)
V = coords.function_space()
du = argument(V)
cds = {coords: du}
if coefficient_derivatives is not None:
cds.update(coefficient_derivatives)
coefficient_derivatives = cds
elif isinstance(u, firedrake.Function):
V = u.function_space()
du = argument(V)
elif isinstance(u, firedrake.Constant):
if u.ufl_shape != ():
raise ValueError("Real function space of vector elements not supported")
V = firedrake.FunctionSpace(mesh, "Real", 0)
du = argument(V)
else:
raise RuntimeError("Can't compute derivative for form")
return ufl.derivative(form, u, du, coefficient_derivatives)
