def apply_matvec(self, A, b, symmetric):
b.allocate(A, dim=0)
for i in range(len(self)):
for bc in self[i]:
for j in range(len(self)):
if i == j:
if numpy.isscalar(A[i, i]):
# Convert to a diagonal matrix, so that the individual rows can be modified
import block.algebraic
A[i, i] = block.algebraic.active_backend(
).create_identity(b[i], val=A[i, i])
if symmetric:
bc.zero_columns(A[i, i], b[i], self.signs[i])
else:
bc.apply(A[i, i], b[i])
else:
if numpy.isscalar(A[i, j]):
if A[i, j] != 0:
dolfin.error(
"can't modify block (%d,%d) for BC, expected a GenericMatrix"
% (i, j))
continue
bc.zero(A[i, j])
if symmetric:
bc.zero_columns(A[j, i], b[j])
def expr2function(self, expr, function):
""" Convert an expression into a function. How this is done is
determined by the parameters (assemble, project or interpolate).
"""
space = function.function_space()
if self.params.expr2function == "assemble":
# Compute average values of expr for each cell and place in a DG0 space
test = TestFunction(space)
scale = 1.0 / CellVolume(space.mesh())
assemble(scale * inner(expr, test) * dx, tensor=function.vector())
return function
elif self.params.expr2function == "project":
# TODO: Avoid superfluous function creation with fenics-dev/1.5 by using:
#project(expr, space, function=function)
function.assign(project(expr, space))
return function
elif self.params.expr2function == "interpolate":
# TODO: Need interpolation with code generated from expr, waiting for uflacs work.
function.interpolate(
expr) # Currently only works if expr is a single Function
return function
else:
error(
"No action selected, need to choose either assemble, project or interpolate."
)
def _action_save(self, field, data, timestep, t):
"Apply the 'save' action to computed field data."
field_name = field.name
# Create save folder first time
self._create_savedir(field_name)
# Collect metadata shared between data types
metadata = {
'timestep': timestep,
'time': t,
}
# Rename Functions to get the right name in file
# (NB! This has the obvious side effect!)
# TODO: We don't need to cache a distinct Function
# object like we do for plotting, or?
if isinstance(data, Function):
data.rename(field_name, "Function produced by cbcpost.")
# Get list of file formats
save_as = _get_save_formats(field, data)
# Write data to file for each filetype
for saveformat in save_as:
# Write data to file depending on type
if saveformat == 'pvd':
metadata[saveformat] = self._update_pvd_file(
field_name, saveformat, data, timestep, t)
elif saveformat == 'xdmf':
metadata[saveformat] = self._update_xdmf_file(
field_name, saveformat, data, timestep, t)
elif saveformat == 'xml':
metadata[saveformat] = self._update_xml_file(
field_name, saveformat, data, timestep, t)
elif saveformat == 'xml.gz':
metadata[saveformat] = self._update_xml_gz_file(
field_name, saveformat, data, timestep, t)
elif saveformat == 'txt':
metadata[saveformat] = self._update_txt_file(
field_name, saveformat, data, timestep, t)
elif saveformat == 'hdf5':
metadata[saveformat] = self._update_hdf5_file(
field_name, saveformat, data, timestep, t)
elif saveformat == 'shelve':
metadata[saveformat] = self._update_shelve_file(
field_name, saveformat, data, timestep, t)
else:
error("Unknown save format %s." % (saveformat, ))
self._timer.completed("PP: save %s %s" % (field_name, saveformat))
# Write new data to metadata file
self._update_metadata_file(field_name, data, t, timestep, save_as,
metadata)
self._timer.completed("PP: update metadata")
self._fill_playlog(field, timestep, save_as)
self._timer.completed("PP: update playlog")
def __init__(self, u, boundary_is_streamline=False, degree=1):
"""
Heavily based on
https://github.com/mikaem/fenicstools/blob/master/fenicstools/Streamfunctions.py
Stream function for a given general 2D velocity field.
The boundary conditions are weakly imposed through the term
inner(q, grad(psi)*n)*ds,
where grad(psi) = [-v, u] is set on all boundaries.
This should work for any collection of boundaries:
walls, inlets, outlets etc.
"""
Vu = u[0].function_space()
mesh = Vu.mesh()
# Check dimension
if not mesh.geometry().dim() == 2:
df.error("Stream-function can only be computed in 2D.")
# Define the weak form
V = df.FunctionSpace(mesh, 'CG', degree)
q = df.TestFunction(V)
psi = df.TrialFunction(V)
n = df.FacetNormal(mesh)
a = df.dot(df.grad(q), df.grad(psi)) * df.dx
L = df.dot(q, df.curl(u)) * df.dx
if boundary_is_streamline:
# Strongly set psi = 0 on entire domain boundary
self.bcs = [df.DirichletBC(V, df.Constant(0), df.DomainBoundary())]
self.normalize = False
else:
self.bcs = []
self.normalize = True
L = L + q * (n[1] * u[0] - n[0] * u[1]) * df.ds
# Create preconditioned iterative solver
solver = df.PETScKrylovSolver('gmres', 'hypre_amg')
solver.parameters['nonzero_initial_guess'] = True
solver.parameters['relative_tolerance'] = 1e-10
solver.parameters['absolute_tolerance'] = 1e-10
# Store for later computation
self.psi = df.Function(V)
self.A = df.assemble(a)
self.L = L
self.mesh = mesh
self.solver = solver
self._triangulation = None
def create_dirichlet_conditions(values, boundaries, function_space):
"""Create Dirichlet boundary conditions for given boundary values,
boundaries and function space."""
# Check that the size matches
if len(values) != len(boundaries):
error("The number of Dirichlet values does not match the number of Dirichlet boundaries.")
info("Creating %d Dirichlet boundary condition(s)." % len(values))
# Create Dirichlet conditions
bcs = []
for (i, value) in enumerate(values):
# Get current boundary
boundary = boundaries[i]
if isinstance(value, tuple):
subdim = value[1]
value = value[0]
temp_function_space = function_space.sub(subdim)
else:
temp_function_space = function_space
# Case 0: boundary is a string
if isinstance(boundary, str):
boundary = CompiledSubDomain(boundary)
bc = DirichletBC(temp_function_space, value, boundary)
# Case 1: boundary is a SubDomain
elif isinstance(boundary, SubDomain):
bc = DirichletBC(temp_function_space, value, boundary)
# Case 2: boundary is defined by a MeshFunction
elif isinstance(boundary, tuple):
mesh_function, index = boundary
bc = DirichletBC(temp_function_space, value, mesh_function, index)
# Unhandled case
else:
error("Unhandled boundary specification for boundary condition. "
"Expecting a string, a SubDomain or a (MeshFunction, int) tuple.")
bcs.append(bc)
return bcs
def __init__(self, parameters=None, solver="ipcs"):
"Create Navier-Stokes problem"
self.parameters = Parameters("problem_parameters")
# Create solver
if solver == "taylor-hood":
info("Using Taylor-Hood based Navier-Stokes solver")
self.solver = TaylorHoodSolver(self)
elif solver == "ipcs":
info("Using IPCS based Navier-Stokes solver")
self.solver = NavierStokesSolver(self)
else:
error("Unknown Navier--Stokes solver: %s" % solver)
# Set up parameters
self.parameters.add(self.solver.parameters)
def __init__(self, parameters=None, solver="ipcs"):
"Create Navier-Stokes problem"
self.parameters = Parameters("problem_parameters")
# Create solver
if solver == "taylor-hood":
info("Using Taylor-Hood based Navier-Stokes solver")
self.solver = TaylorHoodSolver(self)
elif solver == "ipcs":
info("Using IPCS based Navier-Stokes solver")
self.solver = NavierStokesSolver(self)
else:
error("Unknown Navier--Stokes solver: %s" % solver)
# Set up parameters
self.parameters.add(self.solver.parameters)
def create_dirichlet_conditions(values, boundaries, function_spaces):
"""Create Dirichlet boundary conditions for given boundary values,
boundaries and function space."""
# Check that the size matches
if len(values) != len(boundaries):
error(
"The number of Dirichlet values does not match the number of Dirichlet boundaries."
)
if len(values) != len(function_spaces):
error(
"The number of Dirichlet values does not match the number of function spaces."
)
if len(boundaries) != len(function_spaces):
error(
"The number of Dirichlet boundaries does not match the number of function spaces."
)
info("Creating %d Dirichlet boundary condition(s)." % len(values))
# Create Dirichlet conditions
bcs = []
for (i, value) in enumerate(values):
# Get current boundary
boundary = boundaries[i]
function_space = function_spaces[i]
# Case 0: boundary is a string
if isinstance(boundary, str):
boundary = CompiledSubDomain(boundary)
bc = DirichletBC(function_space, value, boundary)
# Case 1: boundary is a SubDomain
elif isinstance(boundary, SubDomain):
bc = DirichletBC(function_space, value, boundary)
# Case 2: boundary is defined by a MeshFunction
elif isinstance(boundary, tuple):
mesh_function, index = boundary
bc = DirichletBC(function_space, value, mesh_function, index)
# Unhandled case
else:
error(
"Unhandled boundary specification for boundary condition. "
"Expecting a string, a SubDomain or a (MeshFunction, int) tuple."
)
bcs.append(bc)
return bcs
def compute_regular_timesteps(problem):
"""Compute fixed timesteps for problem.
The first timestep will be T0 while the last timestep will be in the interval [T, T+dt).
Returns (dt, timesteps, start_timestep).
"""
# Get the time range and timestep from the problem
T0 = problem.params.T0
T = problem.params.T
dt = problem.params.dt
start_timestep = problem.params.start_timestep
# Compute regular timesteps, including T0 and T
timesteps = arange(T0, T+dt, dt)
if abs(dt - (timesteps[1]-timesteps[0])) > 1e-8:
error("Computed timestep size does not match specified dt.")
if timesteps[-1] < T - dt*1e-6:
error("Computed timesteps range does not include end time.")
if timesteps[-1] > T + dt*1e-6:
cbc_warning("End time for simulation does not match end time set for problem (T-T0 not a multiple of dt).")
if start_timestep < 0 or start_timestep >= len(timesteps):
error("start_timestep is beyond the computed timesteps.")
return dt, timesteps, start_timestep
def apply_matvec(self, A, b, symmetric):
b.allocate(A, dim=0)
for i in range(len(self)):
for bc in self[i]:
for j in range(len(self)):
if i==j:
if numpy.isscalar(A[i,i]):
# Convert to a diagonal matrix, so that the individual rows can be modified
import block.algebraic
A[i,i] = block.algebraic.active_backend().create_identity(b[i], val=A[i,i])
if symmetric:
bc.zero_columns(A[i,i], b[i], self.signs[i])
else:
bc.apply(A[i,i], b[i])
else:
if numpy.isscalar(A[i,j]):
if A[i,j] != 0:
dolfin.error("can't modify block (%d,%d) for BC, expected a GenericMatrix" % (i,j))
continue
bc.zero(A[i,j])
if symmetric:
bc.zero_columns(A[j,i], b[j])
def set_initial_conditions(self, **init):
"Update initial_conditions in model"
for init_name, init_value in init.items():
if init_name not in self._initial_conditions:
error("'%s' is not a parameter in %s" %(init_name, self))
if not isinstance(init_value, (float, int)) and not isinstance(init_value._cpp_object, GenericFunction):
error("'%s' is not a scalar or a GenericFunction" % init_name)
if hasattr(init_value, "_cpp_object") and isinstance(init_value._cpp_object, GenericFunction) and \
init_value._cpp_object.value_size() != 1:
error("expected the value_size of '%s' to be 1" % init_name)
self._initial_conditions[init_name] = init_value
def set_parameters(self, **params):
"Update parameters in model"
for param_name, param_value in params.items():
if param_name not in self._parameters:
error("'%s' is not a parameter in %s" %(param_name, self))
if (not isinstance(param_value, (float, int))
and not isinstance(param_value._cpp_object, GenericFunction)):
error("'%s' is not a scalar or a GenericFunction" % param_name)
if hasattr(param_value, "_cpp_object") and\
isinstance(param_value._cpp_object, GenericFunction) and \
param_value._cpp_object.value_size() != 1:
error("expected the value_size of '%s' to be 1" % param_name)
self._parameters[param_name] = param_value
def space_from_string(space_string: str, mesh: df.Mesh,
dim: int) -> df.FunctionSpace:
"""
Constructed a finite elements space from a string
representation of the space
Arguments
---------
space_string : str
A string on the form {familiy}_{degree} which
determines the space. Example 'Lagrange_1'.
mesh : dolfin.Mesh
The mesh
dim : int
1 for scalar space, 3 for vector space.
"""
family, degree = space_string.split("_")
if dim == 3:
V = df.FunctionSpace(
mesh,
df.VectorElement(
family=family,
cell=mesh.ufl_cell(),
degree=int(degree),
quad_scheme="default",
),
)
elif dim == 1:
V = df.FunctionSpace(
mesh,
df.FiniteElement(
family=family,
cell=mesh.ufl_cell(),
degree=int(degree),
quad_scheme="default",
),
)
else:
raise df.error("Cannot create function space of dimension {dim}")
return V
def conductivities(self):
error("Need to prescribe conducitivites")
def mesh(self):
error("Need to prescribe domain")
def scalar_laplacians(mesh):
"""
Calculate the laplacians needed by fiberrule algorithms
Arguments
---------
mesh : dolfin.Mesh
A dolfin mesh with marked boundaries:
base = 10, rv = 20, lv = 30, epi = 40
The base is assumed placed at x=0
"""
if not isinstance(mesh, d.Mesh):
raise TypeError("Expected a dolfin.Mesh as the mesh argument.")
# Init connectivities
mesh.init(2)
facet_markers = d.MeshFunction("size_t", mesh, 2, mesh.domains())
# Boundary markers, solutions and cases
markers = dict(base=10, rv=20, lv=30, epi=40, apex=50)
# Solver parameters
solver_param=dict(solver_parameters=dict(
preconditioner="ml_amg" if d.has_krylov_solver_preconditioner("ml_amg") \
else "default", linear_solver="gmres"))
cases = ["rv", "lv", "epi"]
boundaries = cases + ["base"]
# Check that all boundary faces are marked
num_boundary_facets = d.BoundaryMesh(mesh, "exterior").num_cells()
if num_boundary_facets != sum(np.sum(\
facet_markers.array()==markers[boundary])\
for boundary in boundaries):
d.error("Not all boundary faces are marked correctly. Make sure all "\
"boundary facets are marked as: base = 10, rv = 20, lv = 30, "\
"epi = 40.")
# Coords and cells
coords = mesh.coordinates()
cells_info = mesh.cells()
# Find apex by solving a laplacian with base solution = 0
# Create Base variational problem
V = d.FunctionSpace(mesh, "CG", 1)
u = d.TrialFunction(V)
v = d.TestFunction(V)
a = d.dot(d.grad(u), d.grad(v)) * d.dx
L = v * d.Constant(1) * d.dx
DBC_10 = d.DirichletBC(V, 1, facet_markers, markers["base"], "topological")
# Create solutions
solutions = dict(
(what, d.Function(V)) for what in markers if what != "base")
d.solve(a == L,
solutions["apex"],
DBC_10,
solver_parameters={"linear_solver": "gmres"})
apex_values = solutions["apex"].vector().array()
apex_values[d.dof_to_vertex_map(V)] = solutions["apex"].vector().array()
ind_apex_max = apex_values.argmax()
apex_coord = coords[ind_apex_max, :]
# Update rhs
L = v * d.Constant(0) * d.dx
d.info(" Apex coord: ({0}, {1}, {2})".format(*apex_coord))
d.info(" Num coords: {0}".format(len(coords)))
d.info(" Num cells: {0}".format(len(cells_info)))
# Calculate volume
volume = 0.0
for cell in d.cells(mesh):
volume += cell.volume()
d.info(" Volume: {0}".format(volume))
d.info("")
# Cases
# =====
#
# 1) base: 1, apex: 0
# 2) lv: 1, rv, epi: 0
# 3) rv: 1, lv, epi: 0
# 4) epi: 1, rv, lv: 0
class ApexDomain(d.SubDomain):
def inside(self, x, on_boundary):
return d.near(x[0], apex_coord[0]) and d.near(x[1], apex_coord[1]) and \
d.near(x[2], apex_coord[2])
apex_domain = ApexDomain()
# Case 1:
Poisson = 1
DBC_11 = d.DirichletBC(V, 0, apex_domain, "pointwise")
# Using Poisson
if Poisson:
d.solve(a == L,
solutions["apex"], [DBC_10, DBC_11],
solver_parameters={"linear_solver": "gmres"})
# Using Eikonal equation
else:
Le = v * d.Constant(1) * d.dx
d.solve(a == Le,
solutions["apex"],
DBC_11,
solver_parameters={"linear_solver": "gmres"})
# Create Eikonal problem
eps = d.Constant(mesh.hmax() / 25)
y = solutions["apex"]
F = d.sqrt(d.inner(d.grad(y), d.grad(y)))*v*d.dx - \
d.Constant(1)*v*d.dx + eps*d.inner(d.grad(y), d.grad(v))*d.dx
d.solve(F == 0,
y,
DBC_11,
solver_parameters={
"linear_solver": "lu",
"newton_solver": {
"relative_tolerance": 1e-5
}
})
# Check that solution of the three last cases all sum to 1.
sol = solutions["apex"].vector().copy()
sol[:] = 0.0
# Iterate over the three different cases
for case in cases:
# Solve linear system
bcs = [d.DirichletBC(V, 1 if what == case else 0, \
facet_markers, markers[what], "topological") \
for what in cases]
def I(self, v, s, time=None):
"Return the ionic current."
error("Must define I = I(v, s)")
def num_states(self):
"""Return number of state variables (in addition to the
membrane potential)."""
error("Must overload num_states")
def F(self, v, s, time=None):
"Return right-hand side for state variable evolution."
error("Must define F = F(v, s)")
def __init__(self, tfml_file,system_name='magma',p_name='Pressure',f_name='Porosity',c_name='c',n_name='n',m_name='m',d_name='d',N_name='N',h_squared_name='h_squared',x0_name='x0'):
"""read the tfml_file and use libspud to populate the internal parameters
c: wavespeed
n: permeability exponent
m: bulk viscosity exponent
d: wave dimension
N: number of collocation points
x0: coordinate wave peak
h_squared: the size of the system in compaction lengths
squared (h/delta)**2
"""
# initialize libspud and extract parameters
libspud.clear_options()
libspud.load_options(tfml_file)
# get model dimension
self.dim = libspud.get_option("/geometry/dimension")
self.system_name = system_name
# get solitary wave parameters
path="/system::"+system_name+"/coefficient::"
scalar_value="/type::Constant/rank::Scalar/value::WholeMesh/constant"
vector_value="/type::Constant/rank::Vector/value::WholeMesh/constant::dim"
c = libspud.get_option(path+c_name+scalar_value)
n = int(libspud.get_option(path+n_name+scalar_value))
m = int(libspud.get_option(path+m_name+scalar_value))
d = float(libspud.get_option(path+d_name+scalar_value))
N = int(libspud.get_option(path+N_name+scalar_value))
self.h = np.sqrt(libspud.get_option(path+h_squared_name+scalar_value))
self.x0 = np.array(libspud.get_option(path+x0_name+vector_value))
self.swave = SolitaryWave(c,n,m,d,N)
self.rmax = self.swave.r[-1]
self.tfml_file = tfml_file
# check that d <= dim
assert (d <= self.dim)
# sort out appropriate index for calculating distance r
if d == 1:
self.index = [self.dim - 1]
else:
self.index = range(0,int(d))
# check that the origin point is the correct dimension
assert (len(self.x0) == self.dim)
#read in information for constructing Function space and dolfin objects
# get the mesh parameters and reconstruct the mesh
meshtype = libspud.get_option("/geometry/mesh/source[0]/name")
if meshtype == 'UnitSquare':
number_cells = libspud.get_option("/geometry/mesh[0]/source[0]/number_cells")
diagonal = libspud.get_option("/geometry/mesh[0]/source[0]/diagonal")
mesh = df.UnitSquareMesh(number_cells[0],number_cells[1],diagonal)
elif meshtype == 'Rectangle':
x0 = libspud.get_option("/geometry/mesh::Mesh/source::Rectangle/lower_left")
x1 = libspud.get_option("/geometry/mesh::Mesh/source::Rectangle/upper_right")
number_cells = libspud.get_option("/geometry/mesh::Mesh/source::Rectangle/number_cells")
diagonal = libspud.get_option("/geometry/mesh[0]/source[0]/diagonal")
mesh = df.RectangleMesh(x0[0],x0[1],x1[0],x1[1],number_cells[0],number_cells[1],diagonal)
elif meshtype == 'UnitCube':
number_cells = libspud.get_option("/geometry/mesh[0]/source[0]/number_cells")
mesh = df.UnitCubeMesh(number_cells[0],number_cells[1],number_cells[2])
elif meshtype == 'Box':
x0 = libspud.get_option("/geometry/mesh::Mesh/source::Box/lower_back_left")
x1 = libspud.get_option("/geometry/mesh::Mesh/source::Box/upper_front_right")
number_cells = libspud.get_option("/geometry/mesh::Mesh/source::Box/number_cells")
mesh = df.BoxMesh(x0[0],x0[1],x0[2],x1[0],x1[1],x1[2],number_cells[0],number_cells[1],number_cells[2])
elif meshtype == 'UnitInterval':
number_cells = libspud.get_option("/geometry/mesh::Mesh/source::UnitInterval/number_cells")
mesh = df.UnitIntervalMesh(number_cells)
elif meshtype == 'Interval':
number_cells = libspud.get_option("/geometry/mesh::Mesh/source::Interval/number_cells")
left = libspud.get_option("/geometry/mesh::Mesh/source::Interval/left")
right = libspud.get_option("/geometry/mesh::Mesh/source::Interval/right")
mesh = df.IntervalMesh(number_cells,left,right)
elif meshtype == 'File':
mesh_filename = libspud.get_option("/geometry/mesh::Mesh/source::File/file")
print "tfml_file = ",self.tfml_file, "mesh_filename=",mesh_filename
mesh = df.Mesh(mesh_filename)
else:
df.error("Error: unknown mesh type "+meshtype)
#set the functionspace for n-d solitary waves
path="/system::"+system_name+"/field::"
p_family = libspud.get_option(path+p_name+"/type/rank/element/family")
p_degree = libspud.get_option(path+p_name+"/type/rank/element/degree")
f_family = libspud.get_option(path+f_name+"/type/rank/element/family")
f_degree = libspud.get_option(path+f_name+"/type/rank/element/degree")
pe = df.FiniteElement(p_family, mesh.ufl_cell(), p_degree)
ve = df.FiniteElement(f_family, mesh.ufl_cell(), f_degree)
e = pe*ve
self.functionspace = df.FunctionSpace(mesh, e)
#work out the order of the fields
for i in xrange(libspud.option_count("/system::"+system_name+"/field")):
name = libspud.get_option("/system::"+system_name+"/field["+`i`+"]/name")
if name == f_name:
self.f_index = i
if name == p_name:
self.p_index = i
def I(self, v, s, time=None, index=None):
if index is None:
error("(Domain) index must be specified for multi cell models")
# Extract which cell model index (given by index in incoming tuple)
k = self._key_to_cell_model[index]
return self._cell_models[k].I(v, s, time)
def solve():
error("solve() function not implemented by solver.")
def F(v, s):
error("Must define F = F(v, s)")
def solve():
error("solve() function not implemented by solver.")
def mesh(self):
error("Need to prescribe domain")
def __str__():
error("__str__ not implemented by solver.")
def conductivities(self):
error("Need to prescribe conducitivites")
def I(v, s):
error("Must define I = I(v, s)")
def scalar_laplacians(mesh, markers=None, ffun=None):
"""
Calculate the laplacians
Arguments
---------
mesh : dolfin.Mesh
A dolfin mesh
markers : dict (optional)
A dictionary with the markers for the
different bondaries defined in the facet function
or within the mesh itself.
The follwing markers must be provided:
'base', 'lv', 'epi, 'rv' (optional).
If the markers are not provided the following default
vales will be used: base = 10, rv = 20, lv = 30, epi = 40.
fiber_space : str
A string on the form {familiy}_{degree} which
determines for what space the fibers should be calculated for.
"""
if not isinstance(mesh, df.Mesh):
raise TypeError("Expected a dolfin.Mesh as the mesh argument.")
# Init connectivities
mesh.init(2)
if ffun is None:
ffun = df.MeshFunction("size_t", mesh, 2, mesh.domains())
# Boundary markers, solutions and cases
if markers is None:
markers = utils.default_markers()
else:
keys = ['base', 'lv', 'epi']
msg = ('Key {key} not found in markers. Make sure to provide a'
'key-value pair for {keys}')
for key in keys:
assert key in markers, msg.format(key=key, keys=keys)
if 'rv' not in markers:
df.info(
'No marker for the RV found. Asssume this is an LV geometry')
rv_value = 20
# Just make sure that this value is not used for any of the other boundaries.
while rv_value in markers.values():
rv_value += 1
markers['rv'] = rv_value
markers_str = '\n'.join(
['{}: {}'.format(k, v) for k, v in markers.items()])
df.info(('Compute scalar laplacian solutions with the markers: \n'
'{}').format(markers_str))
cases = ["rv", "lv", "epi"]
boundaries = cases + ["base"]
# Check that all boundary faces are marked
num_boundary_facets = df.BoundaryMesh(mesh, "exterior").num_cells()
if num_boundary_facets != sum(
[np.sum(ffun.array() == markers[boundary])
for boundary in boundaries]):
df.error(("Not all boundary faces are marked correctly. Make sure all "
"boundary facets are marked as: {}"
"").format(", ".join(
["{} = {}".format(k, v) for k, v in markers.items()])))
# Compte the apex to base solutons
apex = apex_to_base(mesh, markers["base"], ffun)
# Find the rest of the laplace soltions
V = apex.function_space()
u = df.TrialFunction(V)
v = df.TestFunction(V)
a = df.dot(df.grad(u), df.grad(v)) * df.dx
L = v * df.Constant(0) * df.dx
solutions = dict((what, df.Function(V)) for what in cases)
solutions["apex"] = apex
df.info(" Num coords: {0}".format(mesh.num_vertices()))
df.info(" Num cells: {0}".format(mesh.num_cells()))
solver_param = dict(solver_parameters=dict(
preconditioner="ml_amg" if df.
has_krylov_solver_preconditioner("ml_amg") else "default",
linear_solver="gmres",
))
# Check that solution of the three last cases all sum to 1.
sol = solutions["apex"].vector().copy()
sol[:] = 0.0
if len(ffun.array()[ffun.array() == markers['rv']]) == 0:
# Remove the RV
cases.pop(next(i for i, c in enumerate(cases) if c == 'rv'))
# Iterate over the three different cases
df.info("Solving Laplace equation")
for case in cases:
df.info(" {0} = 1, {1} = 0".format(
case, ", ".join([c for c in cases if c != case])))
# Solve linear system
bcs = [
df.DirichletBC(V, 1 if what == case else 0, ffun, markers[what],
"topological") for what in cases
]
def __str__():
error("__str__ not implemented by solver.")
def runge_kutta_step(A, b, c,
V,
F,
u0,
t, dt,
sympy_dirichlet_bcs=[],
tol=1.0e-10,
verbose=True
):
'''Runge's third-order method for u' = F(u).
'''
# Make sure that the scheme is strictly upper-triangular.
import numpy
import sympy as smp
s = len(b)
A = numpy.array(A)
if numpy.any(abs(A[numpy.triu_indices(s)]) > 1.0e-15):
error('Butcher tableau not upper triangular.')
u = TrialFunction(V)
v = TestFunction(V)
solver_params = {'linear_solver': 'iterative',
'symmetric': True,
'preconditioner': 'hypre_amg',
'krylov_solver': {'relative_tolerance': tol,
'absolute_tolerance': 0.0,
'maximum_iterations': 100,
'monitor_convergence': verbose
}
}
# For the boundary values, see
#
# Intermediate Boundary Conditions for Runge-Kutta Time Integration of
# Initial-Boundary Value Problems,
# D. Pathria,
# <http://www.math.uh.edu/~hjm/june1995/p00379-p00388.pdf>.
#
tt = smp.symbols('t')
BCS = []
# Get boundary conditions and their derivatives.
for k in range(2):
BCS.append([])
for boundary, expr in sympy_dirichlet_bcs:
# Form k-th derivative.
DexprDt = smp.diff(expr, tt, k)
# TODO set degree of expression
BCS[-1].append(DirichletBC(V,
Expression(smp.printing.ccode(DexprDt), t=t + dt),
boundary))
# Use the Constant() syntax to avoid compiling separate expressions for
# different values of dt.
ddt = Constant(dt)
# Compute the stage values.
k = []
for i in range(s):
U = u0
for j in range(i):
U += ddt * A[i][j] * k[j]
L = F(t + c[i] * dt, U, v)
k.append(Function(V))
# Using this bc is somewhat random.
# TODO come up with something better here.
for g in BCS[1]:
g.t = t + c[i] * dt
solve(u * v * dx == L, k[i],
bcs=BCS[1],
solver_parameters=solver_params
)
#plot(k[-1])
#interactive()
# Put it all together.
U = u0
for i in range(s):
U += b[i] * k[i]
theta = Function(V)
for g in BCS[0]:
g.t = t + dt
solve(u * v * dx == (u0 + ddt * U) * v * dx,
theta,
bcs=BCS[0],
solver_parameters=solver_params
)
return theta