def __init__(self,
lagrangian,
bcs_list,
states,
adjoints,
boundaries,
config,
ksp_options,
adjoint_ksp_options,
shape_scalar_product=None,
deformation_space=None):
"""Initializes the ShapeFormHandler object.
Parameters
----------
lagrangian : cashocs._forms.Lagrangian
The Lagrangian corresponding to the shape optimization problem
bcs_list : list[list[dolfin.fem.dirichletbc.DirichletBC]]
list of boundary conditions for the state variables
states : list[dolfin.function.function.Function]
list of state variables
adjoints : list[dolfin.function.function.Function]
list of adjoint variables
boundaries : dolfin.cpp.mesh.MeshFunctionSizet
a MeshFunction for the boundary markers
config : configparser.ConfigParser
the configparser object storing the problems config
ksp_options : list[list[list[str]]]
The list of command line options for the KSP for the
state systems.
adjoint_ksp_options : list[list[list[str]]]
The list of command line options for the KSP for the
adjoint systems.
shape_scalar_product : ufl.form.Form
The weak form of the scalar product used to determine the
shape gradient.
"""
FormHandler.__init__(self, lagrangian, bcs_list, states, adjoints,
config, ksp_options, adjoint_ksp_options)
self.boundaries = boundaries
self.shape_scalar_product = shape_scalar_product
self.degree_estimation = self.config.getboolean('ShapeGradient',
'degree_estimation',
fallback=False)
self.use_pull_back = self.config.getboolean('ShapeGradient',
'use_pull_back',
fallback=True)
if deformation_space is None:
self.deformation_space = fenics.VectorFunctionSpace(
self.mesh, 'CG', 1)
else:
self.deformation_space = deformation_space
self.test_vector_field = fenics.TestFunction(self.deformation_space)
self.regularization = Regularization(self)
# Calculate the necessary UFL forms
self.inhomogeneous_mu = False
self.__compute_shape_derivative()
self.__compute_shape_gradient_forms()
self.__setup_mu_computation()
if self.degree_estimation:
self.estimated_degree = np.maximum(
estimate_total_polynomial_degree(self.riesz_scalar_product),
estimate_total_polynomial_degree(self.shape_derivative))
self.assembler = fenics.SystemAssembler(self.riesz_scalar_product,
self.shape_derivative,
self.bcs_shape,
form_compiler_parameters={
'quadrature_degree':
self.estimated_degree
})
else:
try:
self.assembler = fenics.SystemAssembler(
self.riesz_scalar_product, self.shape_derivative,
self.bcs_shape)
except (AssertionError, ValueError):
self.estimated_degree = np.maximum(
estimate_total_polynomial_degree(
self.riesz_scalar_product),
estimate_total_polynomial_degree(self.shape_derivative))
self.assembler = fenics.SystemAssembler(
self.riesz_scalar_product,
self.shape_derivative,
self.bcs_shape,
form_compiler_parameters={
'quadrature_degree': self.estimated_degree
})
self.assembler.keep_diagonal = True
self.fe_scalar_product_matrix = fenics.PETScMatrix()
self.fe_shape_derivative_vector = fenics.PETScVector()
self.update_scalar_product()
# test for symmetry
if not self.scalar_product_matrix.isSymmetric():
if not self.scalar_product_matrix.isSymmetric(1e-15):
if not (self.scalar_product_matrix -
self.scalar_product_matrix.copy().transpose()
).norm() / self.scalar_product_matrix.norm() < 1e-15:
raise InputError(
'cashocs._forms.ShapeFormHandler',
'shape_scalar_product',
'Supplied scalar product form is not symmetric.')
if self.opt_algo == 'newton' \
or (self.opt_algo == 'pdas' and self.inner_pdas == 'newton'):
raise NotImplementedError(
'Second order methods are not implemented for shape optimization yet'
)
def damped_newton_solve(F, u, bcs, rtol=1e-10, atol=1e-10, max_iter=50, convergence_type='combined', norm_type='l2',
damped=True, verbose=True, ksp=None, ksp_options=None):
r"""A damped Newton method for solving nonlinear equations.
The damped Newton method is based on the natural monotonicity test from
`Deuflhard, Newton methods for nonlinear problems <https://doi.org/10.1007/978-3-642-23899-4>`_.
It also allows fine tuning via a direct interface, and absolute, relative,
and combined stopping criteria. Can also be used to specify the solver for
the inner (linear) subproblems via petsc ksps.
The method terminates after ``max_iter`` iterations, or if a termination criterion is
satisfied. These criteria are given by
- a relative one in case ``convergence_type = 'rel'``, i.e.,
.. math:: \lvert\lvert F_{k} \rvert\rvert \leq \texttt{rtol} \lvert\lvert F_0 \rvert\rvert.
- an absolute one in case ``convergence_type = 'abs'``, i.e.,
.. math:: \lvert\lvert F_{k} \rvert\rvert \leq \texttt{atol}.
- a combination of both in case ``convergence_type = 'combined'``, i.e.,
.. math:: \lvert\lvert F_{k} \rvert\rvert \leq \texttt{atol} + \texttt{rtol} \lvert\lvert F_0 \rvert\rvert.
The norm chosen for the termination criterion is specified via ``norm_type``.
Parameters
----------
F : ufl.form.Form
The variational form of the nonlinear problem to be solved by Newton's method.
u : dolfin.function.function.Function
The sought solution / initial guess. It is not assumed that the initial guess
satisfies the Dirichlet boundary conditions, they are applied automatically.
The method overwrites / updates this Function.
bcs : list[dolfin.fem.dirichletbc.DirichletBC]
A list of DirichletBCs for the nonlinear variational problem.
rtol : float, optional
Relative tolerance of the solver if convergence_type is either ``'combined'`` or ``'rel'``
(default is ``rtol = 1e-10``).
atol : float, optional
Absolute tolerance of the solver if convergence_type is either ``'combined'`` or ``'abs'``
(default is ``atol = 1e-10``).
max_iter : int, optional
Maximum number of iterations carried out by the method
(default is ``max_iter = 50``).
convergence_type : {'combined', 'rel', 'abs'}
Determines the type of stopping criterion that is used.
norm_type : {'l2', 'linf'}
Determines which norm is used in the stopping criterion.
damped : bool, optional
If ``True``, then a damping strategy is used. If ``False``, the classical
Newton-Raphson iteration (without damping) is used (default is ``True``).
verbose : bool, optional
If ``True``, prints status of the iteration to the console (default
is ``True``).
ksp : petsc4py.PETSc.KSP, optional
The PETSc ksp object used to solve the inner (linear) problem
if this is ``None`` it uses the direct solver MUMPS (default is
``None``).
ksp_options : list[list[str]]
The list of options for the linear solver.
Returns
-------
dolfin.function.function.Function
The solution of the nonlinear variational problem, if converged.
This overrides the input function u.
Examples
--------
Consider the problem
.. math::
\begin{alignedat}{2}
- \Delta u + u^3 &= 1 \quad &&\text{ in } \Omega=(0,1)^2 \\
u &= 0 \quad &&\text{ on } \Gamma.
\end{alignedat}
This is solved with the code ::
from fenics import *
import cashocs
mesh, _, boundaries, dx, _, _ = cashocs.regular_mesh(25)
V = FunctionSpace(mesh, 'CG', 1)
u = Function(V)
v = TestFunction(V)
F = inner(grad(u), grad(v))*dx + pow(u,3)*v*dx - Constant(1)*v*dx
bcs = cashocs.create_bcs_list(V, Constant(0.0), boundaries, [1,2,3,4])
cashocs.damped_newton_solve(F, u, bcs)
"""
if not convergence_type in ['rel', 'abs', 'combined']:
raise InputError('cashocs.nonlinear_solvers.damped_newton_solve', 'convergence_type', 'Input convergence_type has to be one of \'rel\', \'abs\', or \'combined\'.')
if not norm_type in ['l2', 'linf']:
raise InputError('cashocs.nonlinear_solvers.damped_newton_solve', 'norm_type', 'Input norm_type has to be one of \'l2\' or \'linf\'.')
# create the PETSc ksp
if ksp is None:
if ksp_options is None:
ksp_options = [
['ksp_type', 'preonly'],
['pc_type', 'lu'],
['pc_factor_mat_solver_type', 'mumps'],
['mat_mumps_icntl_24', 1]
]
ksp = PETSc.KSP().create()
_setup_petsc_options([ksp], [ksp_options])
ksp.setFromOptions()
# Calculate the Jacobian.
dF = fenics.derivative(F, u)
# Setup increment and function for monotonicity test
V = u.function_space()
du = fenics.Function(V)
ddu = fenics.Function(V)
u_save = fenics.Function(V)
iterations = 0
[bc.apply(u.vector()) for bc in bcs]
# copy the boundary conditions and homogenize them for the increment
bcs_hom = [fenics.DirichletBC(bc) for bc in bcs]
[bc.homogenize() for bc in bcs_hom]
assembler = fenics.SystemAssembler(dF, -F, bcs_hom)
assembler.keep_diagonal = True
A_fenics = fenics.PETScMatrix()
residual = fenics.PETScVector()
# Compute the initial residual
assembler.assemble(A_fenics, residual)
A_fenics.ident_zeros()
A = fenics.as_backend_type(A_fenics).mat()
b = fenics.as_backend_type(residual).vec()
res_0 = residual.norm(norm_type)
if res_0 == 0.0:
if verbose:
print('Residual vanishes, input is already a solution.')
return u
res = res_0
if verbose:
print('Newton Iteration ' + format(iterations, '2d') + ' - residual (abs): '
+ format(res, '.3e') + ' (tol = ' + format(atol, '.3e') + ') residual (rel): '
+ format(res/res_0, '.3e') + ' (tol = ' + format(rtol, '.3e') + ')')
if convergence_type == 'abs':
tol = atol
elif convergence_type == 'rel':
tol = rtol*res_0
else:
tol = rtol*res_0 + atol
# While loop until termination
while res > tol and iterations < max_iter:
iterations += 1
lmbd = 1.0
breakdown = False
u_save.vector()[:] = u.vector()[:]
# Solve the inner problem
_solve_linear_problem(ksp, A, b, du.vector().vec(), ksp_options)
du.vector().apply('')
# perform backtracking in case damping is used
if damped:
while True:
u.vector()[:] += lmbd*du.vector()[:]
assembler.assemble(residual)
b = fenics.as_backend_type(residual).vec()
_solve_linear_problem(ksp=ksp, b=b, x=ddu.vector().vec(), ksp_options=ksp_options)
ddu.vector().apply('')
if ddu.vector().norm(norm_type)/du.vector().norm(norm_type) <= 1:
break
else:
u.vector()[:] = u_save.vector()[:]
lmbd /= 2
if lmbd < 1e-6:
breakdown = True
break
else:
u.vector()[:] += du.vector()[:]
if breakdown:
raise NotConvergedError('Newton solver (state system)', 'Stepsize for increment too low.')
if iterations == max_iter:
raise NotConvergedError('Newton solver (state system)', 'Maximum number of iterations were exceeded.')
# compute the new residual
assembler.assemble(A_fenics, residual)
A_fenics.ident_zeros()
A = fenics.as_backend_type(A_fenics).mat()
b = fenics.as_backend_type(residual).vec()
[bc.apply(residual) for bc in bcs_hom]
res = residual.norm(norm_type)
if verbose:
print('Newton Iteration ' + format(iterations, '2d') + ' - residual (abs): '
+ format(res, '.3e') + ' (tol = ' + format(atol, '.3e') + ') residual (rel): '
+ format(res/res_0, '.3e') + ' (tol = ' + format(rtol, '.3e') + ')')
if res < tol:
if verbose:
print('\nNewton Solver converged after ' + str(iterations) + ' iterations.\n')
break
return u