import logging
import dolfin as df
import numpy as np
from pySDC.core.problem import Problem
from pySDC.implementations.datatype_classes.fenics_mesh import fenics_mesh, rhs_fenics_mesh
# noinspection PyUnusedLocal
[docs]
class fenics_heat(Problem):
r"""
Example implementing the forced one-dimensional heat equation with Dirichlet boundary conditions
.. math::
\frac{d u}{d t} = \nu \frac{d^2 u}{d x^2} + f
for :math:`x \in \Omega:=[0,1]`, where the forcing term :math:`f` is defined by
.. math::
f(x, t) = -\sin(\pi x) (\sin(t) - \nu \pi^2 \cos(t)).
For initial conditions with constant c and
.. math::
u(x, 0) = \sin(\pi x) + c
the exact solution of the problem is given by
.. math::
u(x, t) = \sin(\pi x)\cos(t) + c.
In this class the problem is implemented in the way that the spatial part is solved using ``FEniCS`` [1]_. Hence, the problem
is reformulated to the *weak formulation*
.. math:
\int_\Omega u_t v\,dx = - \nu \int_\Omega \nabla u \nabla v\,dx + \int_\Omega f v\,dx.
The part containing the forcing term is treated explicitly, where it is interpolated in the function space.
The other part will be treated in an implicit way.
Parameters
----------
c_nvars : int, optional
Spatial resolution, i.e., numbers of degrees of freedom in space.
t0 : float, optional
Starting time.
family : str, optional
Indicates the family of elements used to create the function space
for the trail and test functions. The default is ``'CG'``, which are the class
of Continuous Galerkin, a *synonym* for the Lagrange family of elements, see [2]_.
order : int, optional
Defines the order of the elements in the function space.
refinements : int, optional
Denotes the refinement of the mesh. ``refinements=2`` refines the mesh by factor :math:`2`.
nu : float, optional
Diffusion coefficient :math:`\nu`.
c: float, optional
Constant for the Dirichlet boundary condition :math: `c`
Attributes
----------
V : FunctionSpace
Defines the function space of the trial and test functions.
M : scalar, vector, matrix or higher rank tensor
Denotes the expression :math:`\int_\Omega u_t v\,dx`.
K : scalar, vector, matrix or higher rank tensor
Denotes the expression :math:`- \nu \int_\Omega \nabla u \nabla v\,dx`.
g : Expression
The forcing term :math:`f` in the heat equation.
bc : DirichletBC
Denotes the Dirichlet boundary conditions.
References
----------
.. [1] The FEniCS Project Version 1.5. M. S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg,
C. Richardson, J. Ring, M. E. Rognes, G. N. Wells. Archive of Numerical Software (2015).
.. [2] Automated Solution of Differential Equations by the Finite Element Method. A. Logg, K.-A. Mardal, G. N.
Wells and others. Springer (2012).
"""
dtype_u = fenics_mesh
dtype_f = rhs_fenics_mesh
def __init__(self, c_nvars=128, t0=0.0, family='CG', order=4, refinements=1, nu=0.1, c=0.0):
"""Initialization routine"""
# define the Dirichlet boundary
def Boundary(x, on_boundary):
return on_boundary
# set logger level for FFC and dolfin
logging.getLogger('FFC').setLevel(logging.WARNING)
logging.getLogger('UFL').setLevel(logging.WARNING)
# set solver and form parameters
df.parameters["form_compiler"]["optimize"] = True
df.parameters["form_compiler"]["cpp_optimize"] = True
df.parameters['allow_extrapolation'] = True
# set mesh and refinement (for multilevel)
mesh = df.UnitIntervalMesh(c_nvars)
for _ in range(refinements):
mesh = df.refine(mesh)
# define function space for future reference
self.V = df.FunctionSpace(mesh, family, order)
tmp = df.Function(self.V)
print('DoFs on this level:', len(tmp.vector()[:]))
# invoke super init, passing number of dofs, dtype_u and dtype_f
super(fenics_heat, self).__init__(self.V)
self._makeAttributeAndRegister(
'c_nvars', 't0', 'family', 'order', 'refinements', 'nu', 'c', localVars=locals(), readOnly=True
)
# Stiffness term (Laplace)
u = df.TrialFunction(self.V)
v = df.TestFunction(self.V)
a_K = -1.0 * df.inner(df.nabla_grad(u), self.nu * df.nabla_grad(v)) * df.dx
# Mass term
a_M = u * v * df.dx
self.M = df.assemble(a_M)
self.K = df.assemble(a_K)
# set boundary values
self.bc = df.DirichletBC(self.V, df.Constant(c), Boundary)
self.bc_hom = df.DirichletBC(self.V, df.Constant(0), Boundary)
# set forcing term as expression
self.g = df.Expression(
'-sin(a*x[0]) * (sin(t) - b*a*a*cos(t))',
a=np.pi,
b=self.nu,
t=self.t0,
degree=self.order,
)
[docs]
def solve_system(self, rhs, factor, u0, t):
r"""
Dolfin's linear solver for :math:`(M - factor \cdot A) \vec{u} = \vec{rhs}`.
Parameters
----------
rhs : dtype_f
Right-hand side for the nonlinear system.
factor : float
Abbrev. for the node-to-node stepsize (or any other factor required).
u0 : dtype_u
Initial guess for the iterative solver (not used here so far).
t : float
Current time.
Returns
-------
u : dtype_u
Solution.
"""
b = self.apply_mass_matrix(rhs)
u = self.dtype_u(u0)
T = self.M - factor * self.K
self.bc.apply(T, b.values.vector())
df.solve(T, u.values.vector(), b.values.vector())
return u
def __eval_fexpl(self, u, t):
"""
Helper routine to evaluate the explicit part of the right-hand side.
Parameters
----------
u : dtype_u
Current values of the numerical solution (not used here).
t : float
Current time at which the numerical solution is computed.
Returns
-------
fexpl : dtype_u
Explicit part of the right-hand side.
"""
self.g.t = t
fexpl = self.dtype_u(df.interpolate(self.g, self.V))
return fexpl
def __eval_fimpl(self, u, t):
"""
Helper routine to evaluate the implicit part of the right-hand side.
Parameters
----------
u : dtype_u
Current values of the numerical solution.
t : float
Current time at which the numerical solution is computed.
Returns
-------
fimpl : dtype_u
Explicit part of the right-hand side.
"""
tmp = self.dtype_u(self.V)
self.K.mult(u.values.vector(), tmp.values.vector())
fimpl = self.__invert_mass_matrix(tmp)
return fimpl
[docs]
def eval_f(self, u, t):
"""
Routine to evaluate both parts of the right-hand side of the problem.
Parameters
----------
u : dtype_u
Current values of the numerical solution.
t : float
Current time at which the numerical solution is computed.
Returns
-------
f : dtype_f
The right-hand side divided into two parts.
"""
f = self.dtype_f(self.V)
f.impl = self.__eval_fimpl(u, t)
f.expl = self.__eval_fexpl(u, t)
return f
[docs]
def apply_mass_matrix(self, u):
r"""
Routine to apply mass matrix.
Parameters
----------
u : dtype_u
Current values of the numerical solution.
Returns
-------
me : dtype_u
The product :math:`M \vec{u}`.
"""
me = self.dtype_u(self.V)
self.M.mult(u.values.vector(), me.values.vector())
return me
def __invert_mass_matrix(self, u):
r"""
Helper routine to invert mass matrix.
Parameters
----------
u : dtype_u
Current values of the numerical solution.
Returns
-------
me : dtype_u
The product :math:`M^{-1} \vec{u}`.
"""
me = self.dtype_u(self.V)
b = self.dtype_u(u)
M = self.M
self.bc_hom.apply(M, b.values.vector())
df.solve(M, me.values.vector(), b.values.vector())
return me
[docs]
def u_exact(self, t):
r"""
Routine to compute the exact solution at time :math:`t`.
Parameters
----------
t : float
Time of the exact solution.
Returns
-------
me : dtype_u
Exact solution.
"""
u0 = df.Expression('sin(a*x[0]) * cos(t) + c', c=self.c, a=np.pi, t=t, degree=self.order)
me = self.dtype_u(df.interpolate(u0, self.V), val=self.V)
return me
# noinspection PyUnusedLocal
[docs]
class fenics_heat_mass(fenics_heat):
r"""
Example implementing the forced one-dimensional heat equation with Dirichlet boundary conditions
.. math::
\frac{d u}{d t} = \nu \frac{d^2 u}{d x^2} + f
for :math:`x \in \Omega:=[0,1]`, where the forcing term :math:`f` is defined by
.. math::
f(x, t) = -\sin(\pi x) (\sin(t) - \nu \pi^2 \cos(t)).
For initial conditions with constant c and
.. math::
u(x, 0) = \sin(\pi x) + c
the exact solution of the problem is given by
.. math::
u(x, t) = \sin(\pi x)\cos(t) + c.
In this class the problem is implemented in the way that the spatial part is solved using ``FEniCS`` [1]_. Hence, the problem
is reformulated to the *weak formulation*
.. math:
\int_\Omega u_t v\,dx = - \nu \int_\Omega \nabla u \nabla v\,dx + \int_\Omega f v\,dx.
The forcing term is treated explicitly, and is expressed via the mass matrix resulting from the left-hand side term
:math:`\int_\Omega u_t v\,dx`, and the other part will be treated in an implicit way.
Parameters
----------
c_nvars : int, optional
Spatial resolution, i.e., numbers of degrees of freedom in space.
t0 : float, optional
Starting time.
family : str, optional
Indicates the family of elements used to create the function space
for the trail and test functions. The default is ``'CG'``, which are the class
of Continuous Galerkin, a *synonym* for the Lagrange family of elements, see [2]_.
order : int, optional
Defines the order of the elements in the function space.
refinements : int, optional
Denotes the refinement of the mesh. ``refinements=2`` refines the mesh by factor :math:`2`.
nu : float, optional
Diffusion coefficient :math:`\nu`.
Attributes
----------
V : FunctionSpace
Defines the function space of the trial and test functions.
M : scalar, vector, matrix or higher rank tensor
Denotes the expression :math:`\int_\Omega u_t v\,dx`.
K : scalar, vector, matrix or higher rank tensor
Denotes the expression :math:`- \nu \int_\Omega \nabla u \nabla v\,dx`.
g : Expression
The forcing term :math:`f` in the heat equation.
bc : DirichletBC
Denotes the Dirichlet boundary conditions.
bc_hom : DirichletBC
Denotes the homogeneous Dirichlet boundary conditions, potentially required for fixing the residual
fix_bc_for_residual: boolean
flag to indicate that the residual requires special treatment due to boundary conditions
References
----------
.. [1] The FEniCS Project Version 1.5. M. S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg,
C. Richardson, J. Ring, M. E. Rognes, G. N. Wells. Archive of Numerical Software (2015).
.. [2] Automated Solution of Differential Equations by the Finite Element Method. A. Logg, K.-A. Mardal, G. N.
Wells and others. Springer (2012).
"""
def __init__(self, c_nvars=128, t0=0.0, family='CG', order=4, refinements=1, nu=0.1, c=0.0):
"""Initialization routine"""
super().__init__(c_nvars, t0, family, order, refinements, nu, c)
self.fix_bc_for_residual = True
[docs]
def solve_system(self, rhs, factor, u0, t):
r"""
Dolfin's linear solver for :math:`(M - factor A) \vec{u} = \vec{rhs}`.
Parameters
----------
rhs : dtype_f
Right-hand side for the nonlinear system.
factor : float
Abbrev. for the node-to-node stepsize (or any other factor required).
u0 : dtype_u
Initial guess for the iterative solver (not used here so far).
t : float
Current time.
Returns
-------
u : dtype_u
Solution.
"""
u = self.dtype_u(u0)
T = self.M - factor * self.K
b = self.dtype_u(rhs)
self.bc.apply(T, b.values.vector())
df.solve(T, u.values.vector(), b.values.vector())
return u
[docs]
def eval_f(self, u, t):
"""
Routine to evaluate both parts of the right-hand side.
Parameters
----------
u : dtype_u
Current values of the numerical solution.
t : float
Current time at which the numerical solution is computed.
Returns
-------
f : dtype_f
The right-hand side divided into two parts.
"""
f = self.dtype_f(self.V)
self.K.mult(u.values.vector(), f.impl.values.vector())
self.g.t = t
f.expl = self.dtype_u(df.interpolate(self.g, self.V))
f.expl = self.apply_mass_matrix(f.expl)
return f
[docs]
def fix_residual(self, res):
"""
Applies homogeneous Dirichlet boundary conditions to the residual
Parameters
----------
res : dtype_u
Residual
"""
self.bc_hom.apply(res.values.vector())
return None
# noinspection PyUnusedLocal
[docs]
class fenics_heat_mass_timebc(fenics_heat_mass):
r"""
Example implementing the forced one-dimensional heat equation with time-dependent Dirichlet boundary conditions
.. math::
\frac{d u}{d t} = \nu \frac{d^2 u}{d x^2} + f
for :math:`x \in \Omega:=[0,1]`, where the forcing term :math:`f` is defined by
.. math::
f(x, t) = -\cos(\pi x) (\sin(t) - \nu \pi^2 \cos(t)).
and the boundary conditions are given by
.. math::
u(x, t) = \cos(\pi x)\cos(t).
The exact solution of the problem is given by
.. math::
u(x, t) = \cos(\pi x)\cos(t) + c.
In this class the problem is implemented in the way that the spatial part is solved using ``FEniCS`` [1]_. Hence, the problem
is reformulated to the *weak formulation*
.. math:
\int_\Omega u_t v\,dx = - \nu \int_\Omega \nabla u \nabla v\,dx + \int_\Omega f v\,dx.
The forcing term is treated explicitly, and is expressed via the mass matrix resulting from the left-hand side term
:math:`\int_\Omega u_t v\,dx`, and the other part will be treated in an implicit way.
Parameters
----------
c_nvars : int, optional
Spatial resolution, i.e., numbers of degrees of freedom in space.
t0 : float, optional
Starting time.
family : str, optional
Indicates the family of elements used to create the function space
for the trail and test functions. The default is ``'CG'``, which are the class
of Continuous Galerkin, a *synonym* for the Lagrange family of elements, see [2]_.
order : int, optional
Defines the order of the elements in the function space.
refinements : int, optional
Denotes the refinement of the mesh. ``refinements=2`` refines the mesh by factor :math:`2`.
nu : float, optional
Diffusion coefficient :math:`\nu`.
Attributes
----------
V : FunctionSpace
Defines the function space of the trial and test functions.
M : scalar, vector, matrix or higher rank tensor
Denotes the expression :math:`\int_\Omega u_t v\,dx`.
K : scalar, vector, matrix or higher rank tensor
Denotes the expression :math:`- \nu \int_\Omega \nabla u \nabla v\,dx`.
g : Expression
The forcing term :math:`f` in the heat equation.
bc : DirichletBC
Denotes the time-dependent Dirichlet boundary conditions.
bc_hom : DirichletBC
Denotes the homogeneous Dirichlet boundary conditions, potentially required for fixing the residual
fix_bc_for_residual: boolean
flag to indicate that the residual requires special treatment due to boundary conditions
References
----------
.. [1] The FEniCS Project Version 1.5. M. S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg,
C. Richardson, J. Ring, M. E. Rognes, G. N. Wells. Archive of Numerical Software (2015).
.. [2] Automated Solution of Differential Equations by the Finite Element Method. A. Logg, K.-A. Mardal, G. N.
Wells and others. Springer (2012).
"""
def __init__(self, c_nvars=128, t0=0.0, family='CG', order=4, refinements=1, nu=0.1, c=0.0):
"""Initialization routine"""
# define the Dirichlet boundary
def Boundary(x, on_boundary):
return on_boundary
super().__init__(c_nvars, t0, family, order, refinements, nu, c)
self.u_D = df.Expression('cos(a*x[0]) * cos(t) + c', c=self.c, a=np.pi, t=t0, degree=self.order)
self.bc = df.DirichletBC(self.V, self.u_D, Boundary)
self.bc_hom = df.DirichletBC(self.V, df.Constant(0), Boundary)
# set forcing term as expression
self.g = df.Expression(
'-cos(a*x[0]) * (sin(t) - b*a*a*cos(t))',
a=np.pi,
b=self.nu,
t=self.t0,
degree=self.order,
)
[docs]
def solve_system(self, rhs, factor, u0, t):
r"""
Dolfin's linear solver for :math:`(M - factor A) \vec{u} = \vec{rhs}`.
Parameters
----------
rhs : dtype_f
Right-hand side for the nonlinear system.
factor : float
Abbrev. for the node-to-node stepsize (or any other factor required).
u0 : dtype_u
Initial guess for the iterative solver (not used here so far).
t : float
Current time.
Returns
-------
u : dtype_u
Solution.
"""
u = self.dtype_u(u0)
T = self.M - factor * self.K
b = self.dtype_u(rhs)
self.u_D.t = t
self.bc.apply(T, b.values.vector())
self.bc.apply(b.values.vector())
df.solve(T, u.values.vector(), b.values.vector())
return u
[docs]
def u_exact(self, t):
r"""
Routine to compute the exact solution at time :math:`t`.
Parameters
----------
t : float
Time of the exact solution.
Returns
-------
me : dtype_u
Exact solution.
"""
u0 = df.Expression('cos(a*x[0]) * cos(t) + c', c=self.c, a=np.pi, t=t, degree=self.order)
me = self.dtype_u(df.interpolate(u0, self.V), val=self.V)
return me