def project_gradient_neumann(
f0,
degree=None,
mesh=None,
solver_type='gmres',
preconditioner_type='default'
):
"""Find an approximation to f0 that has the same gradient
The resulting function also satisfies homogeneous Neumann boundary
conditions.
Parameters:
f0: the function to approximate
mesh=None: the mesh on which to approximate it If not provided, the
mesh is extracted from f0.
degree=None: degree of the polynomial approximation. extracted
from f0 if not provided.
solver_type='gmres': The linear solver type to use.
preconditioner_type='default': Preconditioner type to use
"""
if not mesh: mesh = f0.function_space().mesh()
element = f0.ufl_element()
if not degree:
degree = element.degree()
CE = FiniteElement('CG', mesh.ufl_cell(), degree)
CS = FunctionSpace(mesh, CE)
DE = FiniteElement('DG', mesh.ufl_cell(), degree)
DS = FunctionSpace(mesh, DE)
CVE = VectorElement('CG', mesh.ufl_cell(), degree - 1)
CV = FunctionSpace(mesh, CVE)
RE = FiniteElement('R', mesh.ufl_cell(), 0)
R = FunctionSpace(mesh, RE)
CRE = MixedElement([CE, RE])
CR = FunctionSpace(mesh, CRE)
f = fe.project(f0, CS,
solver_type=solver_type,
preconditioner_type=preconditioner_type)
g = fe.project(fe.grad(f), CV,
solver_type=solver_type,
preconditioner_type=preconditioner_type)
lf = fe.project(fe.nabla_div(g), CS,
solver_type=solver_type,
preconditioner_type=preconditioner_type)
tf, tc = TrialFunction(CR)
wf, wc = TestFunctions(CR)
dx = Measure('dx', domain=mesh,
metadata={'quadrature_degree': min(degree, 10)})
a = (fe.dot(fe.grad(tf), fe.grad(wf)) + tc * wf + tf * wc) * dx
L = (f * wc - lf * wf) * dx
igc = Function(CR)
fe.solve(a == L, igc,
solver_parameters={'linear_solver': solver_type,
'preconditioner': preconditioner_type}
)
ig, c = igc.sub(0), igc.sub(1)
igd = fe.project(ig, DS,
solver_type=solver_type,
preconditioner_type=preconditioner_type)
return igd
precice.action_write_iteration_checkpoint()): # write checkpoint
precice.store_checkpoint(u_n, t, n)
read_data = precice.read_data()
# Update the coupling expression with the new read data
precice.update_coupling_expression(coupling_expression, read_data)
dt.assign(np.min([fenics_dt, precice_dt]))
# Compute solution
solve(a == L, u_np1, bcs)
# Dirichlet problem obtains flux from solution and sends flux on boundary to Neumann problem
determine_heat_flux(V_g, u_np1, k, fluxes)
fluxes_y = fluxes.sub(1) # only exchange y component of flux.
precice.write_data(fluxes_y)
precice_dt = precice.advance(dt(0))
if precice.is_action_required(precice.action_read_iteration_checkpoint()
): # roll back to checkpoint
u_cp, t_cp, n_cp = precice.retrieve_checkpoint()
u_n.assign(u_cp)
t = t_cp
n = n_cp
else: # update solution
u_n.assign(u_np1)
t += float(dt)
n += 1
class KSDGSolverPeriodic(KSDGSolver):
default_params = dict(
rho_min = 1e-7,
U_min = 1e-7,
width = 1.0,
rhopen = 10,
Upen = 1,
grhopen = 1,
gUpen = 1,
)
def __init__(
self,
mesh=None,
width=1.0,
dim=1,
nelements=8,
degree=2,
parameters={},
V=(lambda U: U),
U0=None,
rho0=None,
t0=0.0,
debug=False,
solver_type = 'lu',
preconditioner_type = 'default',
periodic=True,
ligands=None
):
"""DG solver for the periodic Keller-Segel PDE system
Keyword parameters:
mesh=None: the mesh on which to solve the problem
width=1.0: the width of the domain
dim=1: # of spatial dimensions.
nelements=8: If mesh is not supplied, one will be
contructed using UnitIntervalMesh, UnitSquareMesh, or
UnitCubeMesh (depending on dim). dim and nelements are not
needed if mesh is supplied.
degree=2: degree of the polynomial approximation
parameters={}: a dict giving the values of scalar parameters of
.V, U0, and rho0 Expressions. This dict needs to also
define numerical parameters that appear in the PDE. Some
of these have defaults:
dim = dim: # of spatial dimensions
sigma: organism movement rate
s: attractant secretion rate
gamma: attractant decay rate
D: attractant diffusion constant
rho_min=10.0**-7: minimum feasible worm density
U_min=10.0**-7: minimum feasible attractant concentration
rhopen=10: penalty for discontinuities in rho
Upen=1: penalty for discontinuities in U
grhopen=1, gUpen=1: penalties for discontinuities in gradients
V=(lambda U: U): a callable taking two numerical arguments, U
and rho, or a single argument, U, and returning a single
number, V, the potential corresponding to U. Use fenics
versions of mathematical functions, e.g. fe.ln, abs,
fe.exp.
U0, rho0: Expressions, Functions, or strs specifying the
initial condition.
t0=0.0: initial time
solver_type='lu'
preconditioner_type='default'
periodic=True: Allowed for compatibility, but ignored
ligands=None: ignored for compatibility
"""
logPERIODIC('creating KSDGSolverPeriodic')
self.args = dict(
mesh=mesh,
width=width,
dim=dim,
nelements=nelements,
degree=degree,
parameters=parameters,
V=V,
U0=U0,
rho0=rho0,
t0=t0,
debug=debug,
solver_type = solver_type,
preconditioner_type = preconditioner_type,
periodic=True,
ligands=ligands
)
self.debug = debug
self.solver_type = solver_type
self.preconditioner_type = preconditioner_type
self.periodic = True
self.params = self.default_params.copy()
#
# Store the original mesh in self.omesh. self.mesh will be the
# corner mesh.
#
if (mesh):
self.omesh = mesh
else:
self.omesh = box_mesh(width=width, dim=dim, nelements=nelements)
self.nelements = nelements
try:
comm = self.omesh.mpi_comm().tompi4py()
except AttributeError:
comm = self.omesh.mpi_comm()
self.lmesh = gather_mesh(self.omesh)
omeshstats = mesh_stats(self.omesh)
logPERIODIC('omeshstats', omeshstats)
self.xmin = omeshstats['xmin']
self.xmax = omeshstats['xmax']
self.xmid = omeshstats['xmid']
self.delta_ = omeshstats['dx']
self.mesh = corner_submesh(self.lmesh)
meshstats = mesh_stats(self.mesh)
logPERIODIC('meshstats', meshstats)
logPERIODIC('self.omesh', self.omesh)
logPERIODIC('self.mesh', self.mesh)
logPERIODIC('self.mesh.mpi_comm().size', self.mesh.mpi_comm().size)
self.nelements = nelements
self.degree = degree
self.dim = self.mesh.geometry().dim()
self.params['dim'] = self.dim
self.params.update(parameters)
#
# Solution spaces and Functions
#
# The solution function space is a vector space with
# 2*(2**dim) elements. The first 2**dim components are even
# and odd parts of rho; These are followed by even and
# odd parts of U. The array self.evenodd identifies even
# and odd components. Each row is a length dim sequence 0s and
# 1s and represnts one component. For instance, if evenodd[i]
# is [0, 1, 0], then component i of the vector space is even
# in dimensions 0 and 2 (x and z conventionally) and off in
# dimension 1 (y).
#
self.symmetries = evenodd_symmetries(self.dim)
self.signs = [fe.as_matrix(np.diagflat(1.0 - 2.0*eo))
for eo in self.symmetries]
self.eomat = evenodd_matrix(self.symmetries)
fss = self.make_function_space()
(self.SE, self.SS, self.VE, self.VS) = [
fss[fs] for fs in ('SE', 'SS', 'VE', 'VS')
]
(self.SE, self.SS, self.VE, self.VS) = self.make_function_space()
self.sol = Function(self.VS) # sol, current soln
logPERIODIC('self.sol', self.sol)
# srhos and sUs are fcuntions defiend on subspaces
self.srhos = self.sol.split()[:2**self.dim]
self.sUs = self.sol.split()[2**self.dim:]
# irhos and iUs are Indexed UFL expressions
self.irhos = fe.split(self.sol)[:2**self.dim]
self.iUs = fe.split(self.sol)[2**self.dim:]
self.wrhos = TestFunctions(self.VS)[: 2**self.dim]
self.wUs = TestFunctions(self.VS)[2**self.dim :]
self.tdsol = TrialFunction(self.VS) # time derivatives
self.tdrhos = fe.split(self.tdsol)[: 2**self.dim]
self.tdUs = fe.split(self.tdsol)[2**self.dim :]
bc_method = 'geometric' if self.dim > 1 else 'pointwise'
rhobcs = [DirichletBC(
self.VS.sub(i),
Constant(0),
FacesDomain(self.mesh, self.symmetries[i]),
method=bc_method
) for i in range(2**self.dim) if np.any(self.symmetries[i] != 0.0)]
Ubcs = [DirichletBC(
self.VS.sub(i + 2**self.dim),
Constant(0),
FacesDomain(self.mesh, self.symmetries[i]),
method=bc_method
) for i in range(2**self.dim) if np.any(self.symmetries[i] != 0.0)]
self.bcs = rhobcs + Ubcs
self.n = FacetNormal(self.mesh)
self.h = CellDiameter(self.mesh)
self.havg = fe.avg(self.h)
self.dx = fe.dx
self.dS = fe.dS
#
# record initial state
#
if not U0:
U0 = Constant(0.0)
if isinstance(U0, ufl.coefficient.Coefficient):
self.U0 = U0
else:
self.U0 = Expression(U0, **self.params,
degree=self.degree, domain=self.mesh)
if not rho0:
rho0 = Constant(0.0)
if isinstance(rho0, ufl.coefficient.Coefficient):
self.rho0 = rho0
else:
self.rho0 = Expression(rho0, **self.params,
degree=self.degree, domain=self.mesh)
try:
V(self.U0, self.rho0)
def realV(U, rho):
return V(U, rho)
except TypeError:
def realV(U, rho):
return V(U)
self.V = realV
self.t0 = t0
#
# initialize state
#
# cache assigners
logPERIODIC('restarting')
self.restart()
logPERIODIC('restart returned')
return(None)
def make_function_space(self,
mesh=None,
dim=None,
degree=None
):
if not mesh: mesh = self.mesh
if not dim: dim = self.dim
if not degree: degree = self.degree
SE = FiniteElement('DG', cellShapes[dim-1], degree)
SS = FunctionSpace(mesh, SE) # scalar space
elements = [SE] * (2*2**self.dim)
VE = MixedElement(elements)
VS = FunctionSpace(mesh, VE) # vector space
logPERIODIC('VS', VS)
return dict(SE=SE, SS=SS, VE=VE, VS=VS)
def restart(self):
logPERIODIC('restart')
self.t = self.t0
U0comps = evenodd_functions(
omesh=self.omesh,
degree=self.degree,
func=self.U0,
evenodd=self.symmetries,
width=self.xmax
)
rho0comps = evenodd_functions(
omesh=self.omesh,
degree=self.degree,
func=self.rho0,
evenodd=self.symmetries,
width=self.xmax
)
coords = gather_dof_coords(rho0comps[0].function_space())
for i in range(2**self.dim):
fe.assign(self.sol.sub(i),
function_interpolate(rho0comps[i],
self.SS,
coords=coords))
fe.assign(self.sol.sub(i + 2**self.dim),
function_interpolate(U0comps[i],
self.SS,
coords=coords))
def setup_problem(self, debug=False):
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
drho_integral = vectotal(
[tdrho*wrho*self.dx for tdrho,wrho in
zip(self.tdrhos, self.wrhos)]
)
dU_integral = vectotal(
[tdU*wU*self.dx
for tdU,wU in zip(self.tdUs, self.wUs)
]
)
self.A = fe.assemble(drho_integral + dU_integral)
for bc in self.bcs:
bc.apply(self.A)
# if self.solver_type == 'lu':
# self.solver = fe.LUSolver(
# self.A,
# )
# self.solver.parameters['reuse_factorization'] = True
# else:
# self.solver = fe.KrylovSolver(
# self.A,
# self.solver_type,
# self.preconditioner_type
# )
self.dsol = Function(self.VS)
self.drhos = self.dsol.split()[: 2**self.dim]
self.dUs = self.dsol.split()[2**self.dim :]
#
# These are the values of rho and U themselves (not their
# symmetrized versions) on all subdomains of the original
# domain.
#
if not hasattr(self, 'rhosds'):
self.rhosds = matmul(self.eomat, self.irhos)
if not hasattr(self, 'Usds'):
self.Usds = matmul(self.eomat, self.iUs)
#
# assemble RHS (for each time point, but compile only once)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.params['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rho_min = self.params['rho_min']
self.rhopen = self.params['rhopen']
self.grhopen = self.params['grhopen']
#
# Compute fluxes on subdomains.
#
self.Vsds = [self.V(Usd, rhosd) for Usd,rhosd in
zip(self.Usds, self.rhosds)]
#
# I may need to adjust the signs of the subdomain vs by
# the symmetries of the combinations
#
self.vsds = [-ufl.grad(Vsd) - (
self.s2*ufl.grad(rhosd)/ufl.max_value(rhosd, self.rho_min)
) for Vsd,rhosd in zip(self.Vsds, self.rhosds)]
self.fluxsds = [vsd * rhosd for vsd,rhosd in
zip(self.vsds, self.rhosds)]
self.vnsds = [ufl.max_value(ufl.dot(vsd, self.n), 0)
for vsd in self.vsds]
self.facet_fluxsds = [(
vnsd('+')*ufl.max_value(rhosd('+'), 0.0) -
vnsd('-')*ufl.max_value(rhosd('-'), 0.0)
) for vnsd,rhosd in zip(self.vnsds, self.rhosds)]
#
# Now combine the subdomain fluxes to get the fluxes for
# the symmetrized functions
#
self.fluxs = matmul((2.0**-self.dim)*self.eomat,
self.fluxsds)
self.facet_fluxs = matmul((2.0**-self.dim)*self.eomat,
self.facet_fluxsds)
self.rho_flux_jump = vectotal(
[-facet_flux*ufl.jump(wrho)*self.dS
for facet_flux,wrho in
zip(self.facet_fluxs, self.wrhos)]
)
self.rho_grad_move = vectotal(
[ufl.dot(flux, ufl.grad(wrho))*self.dx
for flux,wrho in
zip(self.fluxs, self.wrhos)]
)
self.rho_penalty = vectotal(
[-(self.rhopen * self.degree**2 / self.havg) *
ufl.dot(ufl.jump(rho, self.n),
ufl.jump(wrho, self.n)) * self.dS
for rho,wrho in zip(self.irhos, self.wrhos)]
)
self.grho_penalty = vectotal(
[-self.grhopen * self.degree**2 *
(ufl.jump(ufl.grad(rho), self.n) *
ufl.jump(ufl.grad(wrho), self.n)) * self.dS
for rho,wrho in zip(self.irhos, self.wrhos)]
)
self.rho_terms = (
self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty
)
if not hasattr(self, 'U_terms'):
self.U_min = self.params['U_min']
self.gamma = self.params['gamma']
self.s = self.params['s']
self.D = self.params['D']
self.Upen = self.params['Upen']
self.gUpen = self.params['gUpen']
self.U_decay = vectotal(
[-self.gamma * U * wU * self.dx
for U,wU in zip(self.iUs, self.wUs)]
)
self.U_secretion = vectotal(
[self.s * rho * wU * self.dx
for rho, wU in zip(self.irhos, self.wUs)]
)
self.jump_gUw = vectotal(
[self.D * ufl.jump(wU * ufl.grad(U), self.n) * self.dS
for wU, U in zip(self.wUs, self.iUs)
]
)
self.U_diffusion = vectotal(
[-self.D
* ufl.dot(ufl.grad(U), ufl.grad(wU))*self.dx
for U,wU in zip(self.iUs, self.wUs)
]
)
self.U_penalty = vectotal(
[-(self.Upen * self.degree**2 / self.havg)
* ufl.dot(ufl.jump(U, self.n), ufl.jump(wU, self.n))*self.dS
for U,wU in zip(self.iUs, self.wUs)
]
)
self.gU_penalty = vectotal(
[-self.gUpen * self.degree**2 *
ufl.jump(ufl.grad(U), self.n) *
ufl.jump(ufl.grad(wU), self.n) * self.dS
for U,wU in zip(self.iUs, self.wUs)
]
)
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty
)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
# if not hasattr(self, 'JU_terms'):
# self.JU_terms = [fe.derivative(self.all_terms, U)
# for U in self.Us]
# if not hasattr(self, 'Jrho_terms'):
# self.Jrho_terms = [fe.derivative(self.all_terms, rho)
# for rho in self.rhos]
def ddt(self, debug=False):
"""Calculate time derivative of rho and U
Results are left in self.dsol as a two-component vector function.
"""
self.setup_problem(debug)
self.b = fe.assemble(self.all_terms)
for bc in self.bcs:
bc.apply(self.b)
return fe.solve(self.A, self.dsol.vector(), self.b,
self.solver_type)
read_data = precice.read_data()
# Update the coupling expression with the new read data
precice.update_coupling_expression(coupling_expression, read_data)
dt.assign(np.min([fenics_dt, precice_dt]))
# Compute solution u^n+1, use bcs u_D^n+1, u^n and coupling bcs
solve(a == L, u_np1, bcs)
# Write data to preCICE according to which problem is being solved
if problem is ProblemType.DIRICHLET:
# Dirichlet problem reads temperature and writes flux on boundary to Neumann problem
determine_gradient(V_g, u_np1, flux)
flux_x = interpolate(flux.sub(0), W)
precice.write_data(flux_x)
elif problem is ProblemType.NEUMANN:
# Neumann problem reads flux and writes temperature on boundary to Dirichlet problem
precice.write_data(u_np1)
precice_dt = precice.advance(dt(0))
# roll back to checkpoint
if precice.is_action_required(precice.action_read_iteration_checkpoint()):
u_cp, t_cp, n_cp = precice.retrieve_checkpoint()
u_n.assign(u_cp)
t = t_cp
n = n_cp
else: # update solution
u_n.assign(u_np1)
class KSDGSolverVariablePeriodic(KSDGSolverVariable, KSDGSolverPeriodic):
default_params = collections.OrderedDict(
sigma=1.0,
rhomin=1e-7,
Umin=1e-7,
width=1.0,
rhopen=10.0,
Upen=1.0,
grhopen=1.0,
gUpen=1.0,
)
def __init__(self,
mesh=None,
width=1.0,
dim=1,
nelements=8,
degree=2,
parameters={},
param_funcs={},
V=(lambda U, params={}: sum(U)),
U0=[],
rho0=None,
t0=0.0,
debug=False,
solver_type='petsc',
preconditioner_type='default',
periodic=True,
ligands=None):
"""Discontinuous Galerkin solver for the Keller-Segel PDE system
Like KSDGSolverVariable, but with periodic boundary conditions.
"""
logVARIABLE('creating KSDGSolverVariablePeriodic')
if not ligands:
ligands = LigandGroups()
else:
ligands = copy.deepcopy(ligands)
self.args = dict(mesh=mesh,
width=width,
dim=dim,
nelements=nelements,
degree=degree,
parameters=parameters,
param_funcs=param_funcs,
V=V,
U0=U0,
rho0=rho0,
t0=t0,
debug=debug,
solver_type=solver_type,
preconditioner_type=preconditioner_type,
periodic=True,
ligands=ligands)
self.t0 = t0
self.debug = debug
self.solver_type = solver_type
self.preconditioner_type = preconditioner_type
self.periodic = True
self.ligands = ligands
self.nligands = ligands.nligands()
self.init_params(parameters, param_funcs)
if nelements is None:
self.nelements = 8
else:
self.nelements = nelements
if (mesh):
self.omesh = self.mesh = mesh
else:
self.omesh = self.mesh = box_mesh(width=width,
dim=dim,
nelements=self.nelements)
self.nelements = nelements
omeshstats = mesh_stats(self.omesh)
try:
comm = self.omesh.mpi_comm().tompi4py()
except AttributeError:
comm = self.omesh.mpi_comm()
self.lmesh = gather_mesh(self.omesh)
logVARIABLE('omeshstats', omeshstats)
self.xmin = omeshstats['xmin']
self.xmax = omeshstats['xmax']
self.xmid = omeshstats['xmid']
self.delta_ = omeshstats['dx']
if nelements is None:
self.nelements = (self.xmax - self.xmin) / self.delta_
self.mesh = corner_submesh(self.lmesh)
meshstats = mesh_stats(self.mesh)
self.degree = degree
self.dim = self.mesh.geometry().dim()
#
# Solution spaces and Functions
#
self.symmetries = evenodd_symmetries(self.dim)
self.signs = [
fe.as_matrix(np.diagflat(1.0 - 2.0 * eo)) for eo in self.symmetries
]
self.eomat = evenodd_matrix(self.symmetries)
fss = self.make_function_space()
(self.SE, self.SS, self.VE,
self.VS) = [fss[fs] for fs in ('SE', 'SS', 'VE', 'VS')]
logVARIABLE('self.VS', self.VS)
self.sol = Function(self.VS) # sol, current soln
logVARIABLE('self.sol', self.sol)
splitsol = self.sol.split()
self.srhos = splitsol[:2**self.dim]
self.sUs = splitsol[2**self.dim:]
splitsol = list(fe.split(self.sol))
self.irhos = splitsol[:2**self.dim]
self.iUs = splitsol[2**self.dim:]
self.iPs = list(fe.split(self.PSf))
self.iparams = collections.OrderedDict(zip(self.param_names, self.iPs))
self.iligands = copy.deepcopy(self.ligands)
self.iligand_params = ParameterList(
[p for p in self.iligands.params() if p[0] in self.param_numbers])
for k in self.iligand_params.keys():
i = self.param_numbers[k]
self.iligand_params[k] = self.iPs[i]
tfs = list(TestFunctions(self.VS))
self.wrhos, self.wUs = tfs[:2**self.dim], tfs[2**self.dim:]
tfs = list(TrialFunctions(self.VS))
self.tdrhos, self.tdUs = tfs[:2**self.dim], tfs[2**self.dim:]
bc_method = 'geometric' if self.dim > 1 else 'pointwise'
rhobcs = [
DirichletBC(self.VS.sub(i),
Constant(0),
FacesDomain(self.mesh, self.symmetries[i]),
method=bc_method) for i in range(2**self.dim)
if np.any(self.symmetries[i] != 0.0)
]
Ubcs = list(
itertools.chain(*[[
DirichletBC(self.VS.sub(i + (lig + 1) * 2**self.dim),
Constant(0),
FacesDomain(self.mesh, self.symmetries[i]),
method=bc_method) for i in range(2**self.dim)
if np.any(self.symmetries[i] != 0.0)
] for lig in range(self.nligands)]))
self.bcs = rhobcs + Ubcs
self.n = FacetNormal(self.mesh)
self.h = CellDiameter(self.mesh)
self.havg = fe.avg(self.h)
self.dx = fe.dx
self.dS = fe.dS
#
# record initial state
#
if not U0:
U0 = [Constant(0.0)] * self.nligands
self.U0s = [Constant(0.0)] * self.nligands
for i, U0i in enumerate(U0):
if isinstance(U0i, ufl.coefficient.Coefficient):
self.U0s[i] = U0i
else:
self.U0s[i] = Expression(U0i,
**self.params,
degree=self.degree,
domain=self.mesh)
if not rho0:
rho0 = Constant(0.0)
if isinstance(rho0, ufl.coefficient.Coefficient):
self.rho0 = rho0
else:
self.rho0 = Expression(rho0,
**self.params,
degree=self.degree,
domain=self.mesh)
self.set_time(t0)
#
# work out how to call V
#
try:
V(self.U0s, self.rho0, params=self.iparams)
def realV(Us, rho):
return V(Us, rho, params=self.iparams)
except TypeError:
def realV(Us, rho):
return V(Us, self.iparams)
self.V = realV
#
# initialize state
#
self.restart()
return None
def make_function_space(self, mesh=None, dim=None, degree=None):
if not mesh: mesh = self.mesh
if not dim: dim = self.dim
if not degree: degree = self.degree
SE = FiniteElement('DG', cellShapes[dim - 1], degree)
SS = FunctionSpace(mesh, SE) # scalar space
elements = [SE] * ((self.nligands + 1) * 2**self.dim)
VE = MixedElement(elements)
VS = FunctionSpace(mesh, VE) # vector space
return dict(SE=SE, SS=SS, VE=VE, VS=VS)
def restart(self):
logVARIABLE('restart')
self.set_time(self.t0)
U0comps = [None] * self.nligands * 2**self.dim
for i, U0i in enumerate(self.U0s):
eofuncs = evenodd_functions(omesh=self.omesh,
degree=self.degree,
func=U0i,
evenodd=self.symmetries,
width=self.xmax)
U0comps[i * 2**self.dim:(i + 1) * 2**self.dim] = eofuncs
rho0comps = evenodd_functions(omesh=self.omesh,
degree=self.degree,
func=self.rho0,
evenodd=self.symmetries,
width=self.xmax)
coords = gather_dof_coords(rho0comps[0].function_space())
for i in range(2**self.dim):
fe.assign(
self.sol.sub(i),
function_interpolate(rho0comps[i], self.SS, coords=coords))
for i in range(self.nligands * 2**self.dim):
fe.assign(self.sol.sub(i + 2**self.dim),
function_interpolate(U0comps[i], self.SS, coords=coords))
def setup_problem(self, t, debug=False):
self.set_time(t)
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
logVARIABLE('making matrix A')
self.drho_integral = sum([
tdrho * wrho * self.dx
for tdrho, wrho in zip(self.tdrhos, self.wrhos)
])
self.dU_integral = sum(
[tdU * wU * self.dx for tdU, wU in zip(self.tdUs, self.wUs)])
logVARIABLE('assembling A')
self.A = fe.PETScMatrix()
logVARIABLE('self.A', self.A)
fe.assemble(self.drho_integral + self.dU_integral, tensor=self.A)
logVARIABLE('A assembled. Applying BCs')
pA = fe.as_backend_type(self.A).mat()
Adiag = pA.getDiagonal()
logVARIABLE('Adiag.array', Adiag.array)
# self.A = fe.assemble(self.drho_integral + self.dU_integral +
# self.dP_integral)
for bc in self.bcs:
bc.apply(self.A)
Adiag = pA.getDiagonal()
logVARIABLE('Adiag.array', Adiag.array)
self.dsol = Function(self.VS)
dsolsplit = self.dsol.split()
self.drhos, self.dUs = (dsolsplit[:2**self.dim],
dsolsplit[2**self.dim:])
#
# assemble RHS (for each time point, but compile only once)
#
#
# These are the values of rho and U themselves (not their
# symmetrized versions) on all subdomains of the original
# domain.
#
if not hasattr(self, 'rhosds'):
self.rhosds = matmul(self.eomat, self.irhos)
# self.Usds is a list of nligands lists. Sublist i is of
# length 2**dim and lists the value of ligand i on each of the
# 2**dim subdomains.
#
if not hasattr(self, 'Usds'):
self.Usds = [
matmul(self.eomat,
self.iUs[i * 2**self.dim:(i + 1) * 2**self.dim])
for i in range(self.nligands)
]
if not hasattr(self, 'rho_terms'):
logVARIABLE('making rho_terms')
self.sigma = self.iparams['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rhomin = self.iparams['rhomin']
self.rhopen = self.iparams['rhopen']
self.grhopen = self.iparams['grhopen']
#
# Compute fluxes on subdomains.
# Vsds is a list of length 2**dim, the value of V on each
# subdomain.
#
self.Vsds = []
for Usd, rhosd in zip(zip(*self.Usds), self.rhosds):
self.Vsds.append(self.V(Usd, ufl.max_value(rhosd,
self.rhomin)))
self.vsds = [
-ufl.grad(Vsd) -
(self.s2 * ufl.grad(rhosd) / ufl.max_value(rhosd, self.rhomin))
for Vsd, rhosd in zip(self.Vsds, self.rhosds)
]
self.fluxsds = [
vsd * rhosd for vsd, rhosd in zip(self.vsds, self.rhosds)
]
self.vnsds = [
ufl.max_value(ufl.dot(vsd, self.n), 0) for vsd in self.vsds
]
self.facet_fluxsds = [
(vnsd('+') * ufl.max_value(rhosd('+'), 0.0) -
vnsd('-') * ufl.max_value(rhosd('-'), 0.0))
for vnsd, rhosd in zip(self.vnsds, self.rhosds)
]
#
# Now combine the subdomain fluxes to get the fluxes for
# the symmetrized functions
#
self.fluxs = matmul((2.0**-self.dim) * self.eomat, self.fluxsds)
self.facet_fluxs = matmul((2.0**-self.dim) * self.eomat,
self.facet_fluxsds)
self.rho_flux_jump = sum([
-facet_flux * ufl.jump(wrho) * self.dS
for facet_flux, wrho in zip(self.facet_fluxs, self.wrhos)
])
self.rho_grad_move = sum([
ufl.dot(flux, ufl.grad(wrho)) * self.dx
for flux, wrho in zip(self.fluxs, self.wrhos)
])
self.rho_penalty = sum([
-(self.degree**2 / self.havg) *
ufl.dot(ufl.jump(rho, self.n),
ufl.jump(self.rhopen * wrho, self.n)) * self.dS
for rho, wrho in zip(self.irhos, self.wrhos)
])
self.grho_penalty = sum([
self.degree**2 *
(ufl.jump(ufl.grad(rho), self.n) *
ufl.jump(ufl.grad(-self.grhopen * wrho), self.n)) * self.dS
for rho, wrho in zip(self.irhos, self.wrhos)
])
self.rho_terms = (self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty)
logVARIABLE('rho_terms made')
if not hasattr(self, 'U_terms'):
logVARIABLE('making U_terms')
self.Umin = self.iparams['Umin']
self.Upen = self.iparams['Upen']
self.gUpen = self.iparams['gUpen']
self.U_decay = 0.0
self.U_secretion = 0.0
self.jump_gUw = 0.0
self.U_diffusion = 0.0
self.U_penalty = 0.0
self.gU_penalty = 0.0
for j, lig in enumerate(self.iligands.ligands()):
sl = slice(j * 2**self.dim, (j + 1) * 2**self.dim)
self.U_decay += sum([
-lig.gamma * iUi * wUi * self.dx
for iUi, wUi in zip(self.iUs[sl], self.wUs[sl])
])
self.U_secretion += sum([
lig.s * rho * wU * self.dx
for rho, wU in zip(self.irhos, self.wUs[sl])
])
self.jump_gUw += sum([
ufl.jump(lig.D * wU * ufl.grad(U), self.n) * self.dS
for wU, U in zip(self.wUs[sl], self.iUs[sl])
])
self.U_diffusion += sum([
-lig.D * ufl.dot(ufl.grad(U), ufl.grad(wU)) * self.dx
for U, wU in zip(self.iUs[sl], self.wUs[sl])
])
self.U_penalty += sum([
(-self.degree**2 / self.havg) *
ufl.dot(ufl.jump(U, self.n),
ufl.jump(self.Upen * wU, self.n)) * self.dS
for U, wU in zip(self.iUs[sl], self.wUs[sl])
])
self.gU_penalty += sum([
-self.degree**2 * ufl.jump(ufl.grad(U), self.n) *
ufl.jump(ufl.grad(self.gUpen * wU), self.n) * self.dS
for U, wU in zip(self.iUs[sl], self.wUs[sl])
])
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty)
logVARIABLE('U_terms made')
if not hasattr(self, 'all_terms'):
logVARIABLE('making all_terms')
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
logVARIABLE('making J_terms')
self.J_terms = fe.derivative(self.all_terms, self.sol)
def ddt(self, t, debug=False):
"""Calculate time derivative of rho and U
Results are left in self.dsol as a two-component vector function.
"""
self.setup_problem(t, debug=debug)
self.b = fe.assemble(self.all_terms)
for bc in self.bcs:
bc.apply(self.b)
return fe.solve(self.A, self.dsol.vector(), self.b, self.solver_type)
class KSDGSolverVariable(KSDGSolverMultiple):
default_params = collections.OrderedDict(
sigma=1.0,
rhomin=1e-7,
Umin=1e-7,
width=1.0,
rhopen=10.0,
Upen=1.0,
grhopen=1.0,
gUpen=1.0,
)
def __init__(self,
mesh=None,
width=1.0,
dim=1,
nelements=8,
degree=2,
parameters={},
param_funcs={},
V=(lambda U, params={}: sum(U)),
U0=[],
rho0=None,
t0=0.0,
debug=False,
solver_type='petsc',
preconditioner_type='default',
periodic=False,
ligands=None):
"""Discontinuous Galerkin solver for the Keller-Segel PDE system
Keyword parameters:
mesh=None: the mesh on which to solve the problem
width=1.0: the width of the domain
dim=1: # of spatial dimensions.
nelements=8: If mesh is not supplied, one will be
contructed using UnitIntervalMesh, UnitSquareMesh, or
UnitCubeMesh (depending on dim). dim and nelements are not
needed if mesh is supplied.
degree=2: degree of the polynomial approximation
parameters={}: a dict giving the initial values of scalar
parameters of .V, U0, and rho0 Expressions. This dict
needs to also define numerical parameters that appear in
the PDE. Some of these have defaults: dim = dim: # of
spatial dimensions sigma: organism movement rate
rhomin=10.0**-7: minimum feasible worm density
Umin=10.0**-7: minimum feasible attractant concentration
rhopen=10: penalty for discontinuities in rho Upen=1:
penalty for discontinuities in U grhopen=1, gUpen=1:
penalties for discontinuities in gradients nligands=1,
number of ligands.
V=(lambda Us, params={}: sum(Us)): a callable taking two
arguments, Us and rho, or a single argument, Us. Us is a
list of length nligands. rho is a single expression. V
returns a single number, V, the potential corresponding to
Us (and rho). Use ufl versions of mathematical functions,
e.g. ufl.ln, abs, ufl.exp.
rho0: Expressions, Functions, or strs specifying the
initial condition for rho.
U0: a list of nligands Expressions, Functions or strs
specifying the initial conditions for the ligands.
t0=0.0: initial time
solver_type='gmres'
preconditioner_type='default'
ligands=LigandGroups(): ligand list
periodic=False: ignored for compatibility
"""
logVARIABLE('creating KSDGSolverVariable')
if not ligands:
ligands = LigandGroups()
else:
ligands = copy.deepcopy(ligands)
self.args = dict(mesh=mesh,
width=width,
dim=dim,
nelements=nelements,
degree=degree,
parameters=parameters,
param_funcs=param_funcs,
V=V,
U0=U0,
rho0=rho0,
t0=t0,
debug=debug,
solver_type=solver_type,
preconditioner_type=preconditioner_type,
periodic=periodic,
ligands=ligands)
self.t0 = t0
self.debug = debug
self.solver_type = solver_type
self.preconditioner_type = preconditioner_type
self.periodic = False
self.ligands = ligands
self.nligands = ligands.nligands()
self.init_params(parameters, param_funcs)
if (mesh):
self.omesh = self.mesh = mesh
else:
self.omesh = self.mesh = box_mesh(width=width,
dim=dim,
nelements=nelements)
self.nelements = nelements
logVARIABLE('self.mesh', self.mesh)
logVARIABLE('self.mesh.mpi_comm().size', self.mesh.mpi_comm().size)
self.nelements = nelements
self.degree = degree
self.dim = self.mesh.geometry().dim()
#
# Solution spaces and Functions
#
fss = self.make_function_space()
(self.SE, self.SS, self.VE,
self.VS) = [fss[fs] for fs in ('SE', 'SS', 'VE', 'VS')]
logVARIABLE('self.VS', self.VS)
self.sol = Function(self.VS) # sol, current soln
logVARIABLE('self.sol', self.sol)
splitsol = self.sol.split()
self.srho, self.sUs = splitsol[0], splitsol[1:]
splitsol = list(fe.split(self.sol))
self.irho, self.iUs = splitsol[0], splitsol[1:]
self.iPs = list(fe.split(self.PSf))
self.iparams = collections.OrderedDict(zip(self.param_names, self.iPs))
self.iligands = copy.deepcopy(self.ligands)
self.iligand_params = ParameterList(
[p for p in self.iligands.params() if p[0] in self.param_numbers])
for k in self.iligand_params.keys():
i = self.param_numbers[k]
self.iligand_params[k] = self.iPs[i]
tfs = list(TestFunctions(self.VS))
self.wrho, self.wUs = tfs[0], tfs[1:]
tfs = list(TrialFunctions(self.VS))
self.tdrho, self.tdUs = tfs[0], tfs[1:]
self.n = FacetNormal(self.mesh)
self.h = CellDiameter(self.mesh)
self.havg = fe.avg(self.h)
self.dx = fe.dx
# self.dx = fe.dx(metadata={'quadrature_degree': min(degree, 10)})
self.dS = fe.dS
# self.dS = fe.dS(metadata={'quadrature_degree': min(degree, 10)})
#
# record initial state
#
try:
V(self.iUs, self.irho, params=self.iparams)
def realV(Us, rho):
return V(Us, rho, params=self.iparams)
except TypeError:
def realV(Us, rho):
return V(Us, self.iparams)
self.V = realV
if not U0:
U0 = [Constant(0.0)] * self.nligands
self.U0s = [Constant(0.0)] * self.nligands
for i, U0i in enumerate(U0):
if isinstance(U0i, ufl.coefficient.Coefficient):
self.U0s[i] = U0i
else:
self.U0s[i] = Expression(U0i,
**self.params,
degree=self.degree,
domain=self.mesh)
if not rho0:
rho0 = Constant(0.0)
if isinstance(rho0, ufl.coefficient.Coefficient):
self.rho0 = rho0
else:
self.rho0 = Expression(rho0,
**self.params,
degree=self.degree,
domain=self.mesh)
self.set_time(t0)
#
# initialize state
#
self.restart()
return None
def init_params(self, parameters, param_funcs):
"""Initialize parameter attributes from __init__ arguments
The attributes initialized are:
self.params0: a dict giving initial values of all parameters
(not just floats). This is basically a copy of the parameters
argument to __init__, with the insertion of 't' as a new
parameter (always param_names[-1]).
self.param_names: a list of the names of the time-varying
parameters. This is the keys of params0 whose corrsponding
values are of type float. The order is the order of the
parameters in self.PSf.
self.nparams: len(self.param_names)
self.param_numbers: a dict mapping param names to numbers
(ints) in the list param_names and the parameters subspace of
the solution FunctionSpace.
self.param_funcs: a dict whose keys are the param_names and
whose values are functions to determine their values as a
function of time, as explained above. These are copied from
the param_funcs argument of __init__, except that the default
initial value function is filled in for parameters not present
in the argument. Also, the function defined for 't' always
returns t.
self.PSf: a Constant object of dimension self.nparams, holding
the initial values of the parameters.
"""
self.param_names = [
n for n, v in parameters.items() if (type(v) is float and n != 't')
]
self.param_names.append('t')
self.nparams = len(self.param_names)
logVARIABLE('self.param_names', self.param_names)
logVARIABLE('self.nparams', self.nparams)
self.param_numbers = collections.OrderedDict(
zip(self.param_names, itertools.count()))
self.params0 = collections.OrderedDict(parameters)
self.params0['t'] = 0.0
self.param_funcs = param_funcs.copy()
def identity(t, params={}):
return t
self.param_funcs['t'] = identity
for n in self.param_names:
if n not in self.param_funcs:
def value0(t, params={}, v0=self.params0[n]):
return v0
self.param_funcs[n] = value0
self.PSf = Constant([self.params0[n] for n in self.param_names])
return
def set_time(self, t):
self.t = t
params = collections.OrderedDict(
zip(self.param_names, self.PSf.values()))
self.PSf.assign(
Constant([
self.param_funcs[n](t, params=params) for n in self.param_names
]))
logVARIABLE('self.t', self.t)
logVARIABLE(
'collections.OrderedDict(zip(self.param_names, self.PSf.values()))',
collections.OrderedDict(zip(self.param_names, self.PSf.values())))
def make_function_space(self, mesh=None, dim=None, degree=None):
if not mesh: mesh = self.mesh
if not dim: dim = self.dim
if not degree: degree = self.degree
SE = FiniteElement('DG', cellShapes[dim - 1], degree)
SS = FunctionSpace(mesh, SE) # scalar space
elements = [SE] * (self.nligands + 1)
VE = MixedElement(elements)
VS = FunctionSpace(mesh, VE) # vector space
return dict(SE=SE, SS=SS, VE=VE, VS=VS)
def restart(self):
logVARIABLE('restart')
self.set_time(self.t0)
CE = FiniteElement('CG', cellShapes[self.dim - 1], self.degree)
CS = FunctionSpace(self.mesh, CE) # scalar space
coords = gather_dof_coords(CS)
fe.assign(self.sol.sub(0),
function_interpolate(self.rho0, self.SS, coords=coords))
for i, U0i in enumerate(self.U0s):
fe.assign(self.sol.sub(i + 1),
function_interpolate(U0i, self.SS, coords=coords))
def setup_problem(self, t, debug=False):
self.set_time(t)
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
self.drho_integral = self.tdrho * self.wrho * self.dx
self.dU_integral = sum([
tdUi * wUi * self.dx for tdUi, wUi in zip(self.tdUs, self.wUs)
])
logVARIABLE('assembling A')
self.A = PETScMatrix()
logVARIABLE('self.A', self.A)
fe.assemble(self.drho_integral + self.dU_integral, tensor=self.A)
logVARIABLE('A assembled. Applying BCs')
self.dsol = Function(self.VS)
dsolsplit = self.dsol.split()
self.drho, self.dUs = dsolsplit[0], dsolsplit[1:]
#
# assemble RHS (for each time point, but compile only once)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.iparams['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rhomin = self.iparams['rhomin']
self.rhopen = self.iparams['rhopen']
self.grhopen = self.iparams['grhopen']
self.v = -ufl.grad(
self.V(self.iUs, ufl.max_value(self.irho, self.rhomin)) -
(self.s2 * ufl.grad(self.irho) /
ufl.max_value(self.irho, self.rhomin)))
self.flux = self.v * self.irho
self.vn = ufl.max_value(ufl.dot(self.v, self.n), 0)
self.facet_flux = ufl.jump(self.vn * ufl.max_value(self.irho, 0.0))
self.rho_flux_jump = -self.facet_flux * ufl.jump(
self.wrho) * self.dS
self.rho_grad_move = ufl.dot(self.flux, ufl.grad(
self.wrho)) * self.dx
self.rho_penalty = -(
(self.degree**2 / self.havg) *
ufl.dot(ufl.jump(self.irho, self.n),
ufl.jump(self.rhopen * self.wrho, self.n)) * self.dS)
self.grho_penalty = -(
self.degree**2 *
(ufl.jump(ufl.grad(self.irho), self.n) * ufl.jump(
ufl.grad(self.grhopen * self.wrho), self.n)) * self.dS)
self.rho_terms = (self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty)
if not hasattr(self, 'U_terms'):
self.Umin = self.iparams['Umin']
self.Upen = self.iparams['Upen']
self.gUpen = self.iparams['gUpen']
self.U_decay = sum([
-lig.gamma * iUi * wUi * self.dx for lig, iUi, wUi in zip(
self.iligands.ligands(), self.iUs, self.wUs)
])
self.U_secretion = sum([
lig.s * self.irho * wUi * self.dx
for lig, wUi in zip(self.iligands.ligands(), self.wUs)
])
self.jump_gUw = sum([
ufl.jump(lig.D * wUi * ufl.grad(iUi), self.n) * self.dS
for lig, wUi, iUi in zip(self.iligands.ligands(), self.wUs,
self.iUs)
])
self.U_diffusion = sum([
-lig.D * ufl.dot(ufl.grad(iUi), ufl.grad(wUi)) * self.dx
for lig, iUi, wUi in zip(self.iligands.ligands(), self.iUs,
self.wUs)
])
self.U_penalty = sum([
-(self.degree**2 / self.havg) * ufl.dot(
ufl.jump(iUi, self.n), ufl.jump(self.Upen * wUi, self.n)) *
self.dS for iUi, wUi in zip(self.iUs, self.wUs)
])
self.gU_penalty = sum([
-self.degree**2 * ufl.jump(ufl.grad(iUi), self.n) *
ufl.jump(ufl.grad(self.gUpen * wUi), self.n) * self.dS
for iUi, wUi in zip(self.iUs, self.wUs)
])
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
def ddt(self, t, debug=False):
"""Calculate time derivative of rho and U
Results are left in self.dsol as a two-component vector function.
"""
self.setup_problem(t, debug=debug)
self.b = fe.assemble(self.all_terms)
return fe.solve(self.A, self.dsol.vector(), self.b, self.solver_type)
class KSDGSolver:
default_params = dict(
rho_min=1e-7,
U_min=1e-7,
width=1.0,
rhopen=10,
Upen=1,
grhopen=1,
gUpen=1,
)
def __init__(self,
mesh=None,
width=1.0,
dim=1,
nelements=8,
degree=2,
parameters={},
V=(lambda U: U),
U0=None,
rho0=None,
t0=0.0,
debug=False,
solver_type='gmres',
preconditioner_type='default',
periodic=False,
ligands=None):
"""Discontinuous Galerkin solver for the Keller-Segel PDE system
Keyword parameters:
mesh=None: the mesh on which to solve the problem
width=1.0: the width of the domain
dim=1: # of spatial dimensions.
nelements=8: If mesh is not supplied, one will be
contructed using UnitIntervalMesh, UnitSquareMesh, or
UnitCubeMesh (depending on dim). dim and nelements are not
needed if mesh is supplied.
degree=2: degree of the polynomial approximation
parameters={}: a dict giving the values of scalar parameters of
.V, U0, and rho0 Expressions. This dict needs to also
define numerical parameters that appear in the PDE. Some
of these have defaults:
dim = dim: # of spatial dimensions
sigma: organism movement rate
s: attractant secretion rate
gamma: attractant decay rate
D: attractant diffusion constant
rho_min=10.0**-7: minimum feasible worm density
U_min=10.0**-7: minimum feasible attractant concentration
rhopen=10: penalty for discontinuities in rho
Upen=1: penalty for discontinuities in U
grhopen=1, gUpen=1: penalties for discontinuities in gradients
V=(lambda U: U): a callable taking two numerical arguments, U
and rho, or a single argument, U, and returning a single
number, V, the potential corresponding to U. Use fenics
versions of mathematical functions, e.g. ufl.ln, abs,
ufl.exp.
U0, rho0: Expressions, Functions, or strs specifying the
initial condition.
t0=0.0: initial time
solver_type='gmres'
preconditioner_type='default'
periodic, ligands: ignored for caompatibility
"""
logSOLVER('creating KSDGSolver')
self.args = dict(mesh=mesh,
width=width,
dim=dim,
nelements=nelements,
degree=degree,
parameters=parameters,
V=V,
U0=U0,
rho0=rho0,
t0=t0,
debug=debug,
solver_type=solver_type,
preconditioner_type=preconditioner_type,
periodic=periodic,
ligands=ligands)
self.debug = debug
self.solver_type = solver_type
self.preconditioner_type = preconditioner_type
self.periodic = False
self.ligands = ligands
self.params = self.default_params.copy()
if (mesh):
self.omesh = self.mesh = mesh
else:
self.omesh = self.mesh = box_mesh(width=width,
dim=dim,
nelements=nelements)
self.nelements = nelements
logSOLVER('self.mesh', self.mesh)
logSOLVER('self.mesh.mpi_comm().size', self.mesh.mpi_comm().size)
self.nelements = nelements
self.degree = degree
self.dim = self.mesh.geometry().dim()
self.params['dim'] = self.dim
self.params.update(parameters)
#
# Solution spaces and Functions
#
fss = self.make_function_space()
(self.SE, self.SS, self.VE,
self.VS) = [fss[fs] for fs in ('SE', 'SS', 'VE', 'VS')]
logSOLVER('self.VS', self.VS)
self.sol = Function(self.VS) # sol, current soln
logSOLVER('self.sol', self.sol)
self.srho, self.sU = self.sol.sub(0), self.sol.sub(1)
self.irho, self.iU = fe.split(self.sol)
self.wrho, self.wU = TestFunctions(self.VS)
self.tdsol = TrialFunction(self.VS)
self.tdrho, self.tdU = fe.split(self.tdsol)
self.n = FacetNormal(self.mesh)
self.h = CellDiameter(self.mesh)
self.havg = fe.avg(self.h)
self.dx = fe.dx
# self.dx = fe.dx(metadata={'quadrature_degree': min(degree, 10)})
self.dS = fe.dS
# self.dS = fe.dS(metadata={'quadrature_degree': min(degree, 10)})
#
# record initial state
#
try:
V(self.iU, self.irho)
def realV(U, rho):
return V(U, rho)
except TypeError:
def realV(U, rho):
return V(U)
self.V = realV
if not U0:
U0 = Constant(0.0)
if isinstance(U0, ufl.coefficient.Coefficient):
self.U0 = U0
else:
self.U0 = Expression(U0,
**self.params,
degree=self.degree,
domain=self.mesh)
if not rho0:
rho0 = Constant(0.0)
if isinstance(rho0, ufl.coefficient.Coefficient):
self.rho0 = rho0
else:
self.rho0 = Expression(rho0,
**self.params,
degree=self.degree,
domain=self.mesh)
self.t0 = t0
#
# initialize state
#
# cache assigners
logSOLVER('restarting')
self.restart()
logSOLVER('restart returned')
return (None)
def make_function_space(self, mesh=None, dim=None, degree=None):
if not mesh: mesh = self.mesh
if not dim: dim = self.dim
if not degree: degree = self.degree
SE = FiniteElement('DG', cellShapes[dim - 1], degree)
SS = FunctionSpace(mesh, SE) # scalar space
VE = MixedElement(SE, SE)
VS = FunctionSpace(mesh, VE) # vector space
return dict(SE=SE, SS=SS, VE=VE, VS=VS)
def restart(self):
logSOLVER('restart')
self.t = self.t0
CE = FiniteElement('CG', cellShapes[self.dim - 1], self.degree)
CS = FunctionSpace(self.mesh, CE) # scalar space
coords = gather_dof_coords(CS)
logSOLVER('function_interpolate(self.U0, self.SS, coords=coords)',
function_interpolate(self.U0, self.SS, coords=coords))
fe.assign(self.sol.sub(1),
function_interpolate(self.U0, self.SS, coords=coords))
logSOLVER('U0 assign returned')
fe.assign(self.sol.sub(0),
function_interpolate(self.rho0, self.SS, coords=coords))
def set_time(t):
"""Stub for derived classes to override"""
self.t = t
def setup_problem(self, debug=False):
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
self.drho_integral = self.tdrho * self.wrho * self.dx
self.dU_integral = self.tdU * self.wU * self.dx
self.A = fe.assemble(self.drho_integral + self.dU_integral)
# if self.solver_type == 'lu':
# self.solver = fe.LUSolver(
# self.A,
# )
# self.solver.parameters['reuse_factorization'] = True
# else:
# self.solver = fe.KrylovSolver(
# self.A,
# self.solver_type,
# self.preconditioner_type
# )
# self.solver.parameters.add('linear_solver', self.solver_type)
# kparams = fe.Parameters('krylov_solver')
# kparams.add('report', True)
# kparams.add('nonzero_initial_guess', True)
# self.solver.parameters.add(kparams)
# lparams = fe.Parameters('lu_solver')
# lparams.add('report', True)
# lparams.add('reuse_factorization', True)
# lparams.add('verbose', True)
# self.solver.parameters.add(lparams)
self.dsol = Function(self.VS)
self.drho, self.dU = self.dsol.sub(0), self.dsol.sub(1)
#
# assemble RHS (for each time point, but compile only once)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.params['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rho_min = self.params['rho_min']
self.rhopen = self.params['rhopen']
self.grhopen = self.params['grhopen']
self.v = -ufl.grad(self.V(self.iU, self.irho)) - (
self.s2 * ufl.grad(self.irho) /
ufl.max_value(self.irho, self.rho_min))
self.flux = self.v * self.irho
self.vn = ufl.max_value(ufl.dot(self.v, self.n), 0)
self.facet_flux = (
self.vn('+') * ufl.max_value(self.irho('+'), 0.0) -
self.vn('-') * ufl.max_value(self.irho('-'), 0.0))
self.rho_flux_jump = -self.facet_flux * ufl.jump(
self.wrho) * self.dS
self.rho_grad_move = ufl.dot(self.flux, ufl.grad(
self.wrho)) * self.dx
self.rho_penalty = -(
(self.rhopen * self.degree**2 / self.havg) * ufl.dot(
ufl.jump(self.irho, self.n), ufl.jump(self.wrho, self.n)) *
self.dS)
self.grho_penalty = -(self.grhopen * self.degree**2 *
(ufl.jump(ufl.grad(self.irho), self.n) *
ufl.jump(ufl.grad(self.wrho), self.n)) *
self.dS)
self.rho_terms = (self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty)
if not hasattr(self, 'U_terms'):
self.U_min = self.params['U_min']
self.gamma = self.params['gamma']
self.s = self.params['s']
self.D = self.params['D']
self.Upen = self.params['Upen']
self.gUpen = self.params['gUpen']
self.U_decay = -self.gamma * self.iU * self.wU * self.dx
self.U_secretion = self.s * self.irho * self.wU * self.dx
self.jump_gUw = (self.D *
ufl.jump(self.wU * ufl.grad(self.iU), self.n) *
self.dS)
self.U_diffusion = -self.D * ufl.dot(ufl.grad(self.iU),
ufl.grad(self.wU)) * self.dx
self.U_penalty = -(
(self.Upen * self.degree**2 / self.havg) *
ufl.dot(ufl.jump(self.iU, self.n), ufl.jump(self.wU, self.n)) *
self.dS)
self.gU_penalty = -(self.gUpen * self.degree**2 *
(ufl.jump(ufl.grad(self.iU), self.n) *
ufl.jump(ufl.grad(self.wU), self.n)) *
self.dS)
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
# if not hasattr(self, 'JU_terms'):
# self.JU_terms = fe.derivative(self.all_terms, self.sU)
# if not hasattr(self, 'Jrho_terms'):
# self.Jrho_terms = fe.derivative(self.all_terms, self.srho)
def ddt(self, debug=False):
"""Calculate time derivative of rho and U
Results are left in self.dsol as a two-component vector function.
"""
self.setup_problem(debug)
self.b = fe.assemble(self.all_terms)
return fe.solve(self.A, self.dsol.vector(), self.b, self.solver_type)
#
# The following member functions should not really exist -- use
# the TS classes in ts instead. These are here only to allow some
# old code to work.
def implicitTS(self,
t0=0.0,
dt=0.001,
tmax=20,
maxsteps=100,
rtol=1e-5,
atol=1e-5,
prt=True,
restart=True,
tstype=PETSc.TS.Type.ROSW,
finaltime=PETSc.TS.ExactFinalTime.STEPOVER):
"""
Create an implicit timestepper and solve the DE
Keyword arguments:
t0=0.0: the initial time.
dt=0.001: the initial time step.
tmax=20: the final time.
maxsteps=100: maximum number of steps to take.
rtol=1e-5: relative error tolerance.
atol=1e-5: absolute error tolerance.
prt=True: whether to print results as solution progresses
restart=True: whether to set the initial condition to rho0, U0
tstype=PETSc.TS.Type.ROSW: implicit solver to use.
finaltime=PETSc.TS.ExactFinalTime.STEPOVER: how to handle
final time step.
Other options can be set by modifying the PETSc Options
database.
"""
# print("KSDGSolver.implicitTS __init__ entered")
from .ts import implicitTS # done here to avoid circular imports
self.ts = implicitTS(
self,
t0=t0,
dt=dt,
tmax=tmax,
maxsteps=maxsteps,
rtol=rtol,
atol=atol,
restart=restart,
tstype=tstype,
finaltime=finaltime,
)
self.ts.setMonitor(self.ts.historyMonitor)
if prt:
self.ts.setMonitor(self.ts.printMonitor)
self.ts.solve()
self.ts.cleanup()
def cleanupTS():
"""Should be called when finished with a TS
Leaves history unchanged.
"""
del self.ts, self.tsparams
def imExTS(self,
t0=0.0,
dt=0.001,
tmax=20,
maxsteps=100,
rtol=1e-5,
atol=1e-5,
prt=True,
restart=True,
tstype=PETSc.TS.Type.ARKIMEX,
finaltime=PETSc.TS.ExactFinalTime.STEPOVER):
"""
Create an implicit/explicit timestepper and solve the DE
Keyword arguments:
t0=0.0: the initial time.
dt=0.001: the initial time step.
tmax=20: the final time.
maxsteps=100: maximum number of steps to take.
rtol=1e-5: relative error tolerance.
atol=1e-5: absolute error tolerance.
prt=True: whether to print results as solution progresses
restart=True: whether to set the initial condition to rho0, U0
tstype=PETSc.TS.Type.ARKIMEX: implicit solver to use.
finaltime=PETSc.TS.ExactFinalTime.STEPOVER: how to handle
final time step.
Other options can be set by modifying the PETSc options
database.
"""
from .ts import imExTS # done here to avoid circular imports
self.ts = imExTS(
self,
t0=t0,
dt=dt,
tmax=tmax,
maxsteps=maxsteps,
rtol=rtol,
atol=atol,
restart=restart,
tstype=tstype,
finaltime=finaltime,
)
self.ts.setMonitor(self.ts.historyMonitor)
if prt:
self.ts.setMonitor(self.ts.printMonitor)
self.ts.solve()
self.ts.cleanup()
def explicitTS(self,
t0=0.0,
dt=0.001,
tmax=20,
maxsteps=100,
rtol=1e-5,
atol=1e-5,
prt=True,
restart=True,
tstype=PETSc.TS.Type.RK,
finaltime=PETSc.TS.ExactFinalTime.STEPOVER):
"""
Create an explicit timestepper and solve the DE
Keyword arguments:
t0=0.0: the initial time.
dt=0.001: the initial time step.
tmax=20: the final time.
maxsteps=100: maximum number of steps to take.
rtol=1e-5: relative error tolerance.
atol=1e-5: absolute error tolerance.
prt=True: whether to print results as solution progresses
restart=True: whether to set the initial condition to rho0, U0
tstype=PETSc.TS.Type.RK: explicit solver to use.
finaltime=PETSc.TS.ExactFinalTime.STEPOVER: how to handle
final time step.
Other options can be set by modifyign the PETSc options
database.
"""
from .ts import explicitTS # done here to avoid circular imports
self.ts = explicitTS(
self,
t0=t0,
dt=dt,
tmax=tmax,
maxsteps=maxsteps,
rtol=rtol,
atol=atol,
restart=restart,
tstype=tstype,
finaltime=finaltime,
)
self.ts.setMonitor(self.ts.historyMonitor)
if prt:
self.ts.setMonitor(self.ts.printMonitor)
self.ts.solve()
self.ts.cleanup()
class EKKSDGSolver(KSDGSolver):
"""KSDSolver that uses the Epshteyn and Kurganov scheme for rho.
Overrides the setup_problem method of KSDGSolver.
"""
def setup_problem(self, debug=False):
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
drho_integral = self.tdrho * self.wrho * self.dx
dU_integral = self.tdU * self.wU * self.dx
self.A = fe.assemble(drho_integral + dU_integral)
# if self.solver_type == 'lu':
# self.solver = fe.LUSolver(
# self.A,
# method=self.solver_type
# )
# self.solver.parameters['reuse_factorization'] = True
# else:
# self.solver = fe.KrylovSolver(
# self.A,
# self.solver_type,
# self.preconditioner_type
# )
self.dsol = Function(self.VS)
self.drho, self.dU = self.dsol.sub(0), self.dsol.sub(1)
#
# assemble RHS (has to be done for each time point)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.params['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rho_min = self.params['rho_min']
self.rhopen = self.params['rhopen']
self.grhopen = self.params['grhopen']
self.v = -ufl.grad(self.V(self.iU, self.irho))
self.flux = self.v * self.irho
self.vn = ufl.max_value(ufl.dot(self.v, self.n), 0)
self.facet_flux = (self.vn('+') * self.irho('+') -
self.vn('-') * self.irho('-'))
self.rho_flux_jump = -self.facet_flux * ufl.jump(
self.wrho) * self.dS
self.rho_grad_move = ufl.dot(self.flux, ufl.grad(
self.wrho)) * self.dx
self.rho_penalty = -(
(self.rhopen * self.degree**2 / self.havg) * ufl.dot(
ufl.jump(self.irho, self.n), ufl.jump(self.wrho, self.n)) *
self.dS)
# self.facet_flux = (
# self.vn('+')*self.rho('+') - self.vn('-')*self.rho('-')
# )
# self.rho_flux_jump = -self.facet_flux*ufl.jump(self.wrho)*self.dS
# self.rho_grad_move = ufl.dot(self.flux, ufl.grad(self.wrho))*self.dx
self.jump_grhow = (
self.s2 * ufl.jump(self.wrho * ufl.grad(self.irho), self.n) *
self.dS)
self.rho_diffusion = -self.s2 * ufl.dot(ufl.grad(
self.irho), ufl.grad(self.wrho)) * self.dx
# self.rho_penalty = -(
# (self.rhopen * self.degree**2 / self.havg) *
# ufl.dot(ufl.jump(self.rho, self.n),
# ufl.jump(self.wrho, self.n)) * self.dS
# )
self.grho_penalty = -(self.grhopen * self.degree**2 *
(ufl.jump(ufl.grad(self.irho), self.n) *
ufl.jump(ufl.grad(self.wrho), self.n)) *
self.dS)
self.rho_terms = (
# advection terms
self.rho_flux_jump + self.rho_grad_move +
# diffusive terms
self.rho_diffusion + self.jump_grhow +
# penalty terms (to enforce continuity)
self.rho_penalty + self.grho_penalty)
if not hasattr(self, 'U_terms'):
self.U_min = self.params['U_min']
self.gamma = self.params['gamma']
self.s = self.params['s']
self.D = self.params['D']
self.Upen = self.params['Upen']
self.gUpen = self.params['gUpen']
self.U_decay = -self.gamma * self.iU * self.wU * self.dx
self.U_secretion = self.s * self.irho * self.wU * self.dx
self.jump_gUw = (self.D *
ufl.jump(self.wU * ufl.grad(self.iU), self.n) *
self.dS)
self.U_diffusion = -self.D * ufl.dot(ufl.grad(self.iU),
ufl.grad(self.wU)) * self.dx
self.U_penalty = -(
(self.Upen * self.degree**2 / self.havg) *
ufl.dot(ufl.jump(self.iU, self.n), ufl.jump(self.wU, self.n)) *
self.dS)
self.gU_penalty = -(self.gUpen * self.degree**2 *
(ufl.jump(ufl.grad(self.iU), self.n) *
ufl.jump(ufl.grad(self.wU), self.n)) *
self.dS)
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
class KSDGSolverMultiple(KSDGSolver):
default_params = dict(
rho_min = 1e-7,
U_min = 1e-7,
width = 1.0,
rhopen = 10,
Upen = 1,
grhopen = 1,
gUpen = 1,
ligands = None,
)
def __init__(
self,
mesh=None,
width=1.0,
dim=1,
nelements=8,
degree=2,
parameters={},
V=(lambda U: U),
U0=[],
rho0=None,
t0=0.0,
debug=False,
solver_type = 'gmres',
preconditioner_type = 'default',
periodic=False,
ligands=None
):
"""Discontinuous Galerkin solver for the Keller-Segel PDE system
Keyword parameters:
mesh=None: the mesh on which to solve the problem
width=1.0: the width of the domain
dim=1: # of spatial dimensions.
nelements=8: If mesh is not supplied, one will be
contructed using UnitIntervalMesh, UnitSquareMesh, or
UnitCubeMesh (depending on dim). dim and nelements are not
needed if mesh is supplied.
degree=2: degree of the polynomial approximation
parameters={}: a dict giving the values of scalar parameters of
.V, U0, and rho0 Expressions. This dict needs to also
define numerical parameters that appear in the PDE. Some
of these have defaults:
dim = dim: # of spatial dimensions
sigma: organism movement rate
rho_min=10.0**-7: minimum feasible worm density
U_min=10.0**-7: minimum feasible attractant concentration
rhopen=10: penalty for discontinuities in rho
Upen=1: penalty for discontinuities in U
grhopen=1, gUpen=1: penalties for discontinuities in gradients
nligands=1, number of ligands
V=(lambda Us: Us): a callable taking two arguments, Us and
rho, or a single argument, Us. Us is a list of length
nligands. rho is a single expression. V returns a single
number, V, the potential corresponding to Us (and
rho). Use ufl versions of mathematical functions,
e.g. ufl.ln, abs, ufl.exp.
rho0: Expressions, Functions, or strs specifying the
initial condition for rho.
U0: a list of nligands Expressions, Functions or strs
specifying the initial conditions for the ligands.
t0=0.0: initial time
solver_type='gmres'
preconditioner_type='default'
ligands=LigandGroups(): ligand list
periodic=False: ignored for compatibility
"""
logMULTIPLE('creating KSDGSolverMultiple')
if not ligands:
ligands = LigandGroups()
else:
ligands = copy.deepcopy(ligands)
self.args = dict(
mesh=mesh,
width=width,
dim=dim,
nelements=nelements,
degree=degree,
parameters=parameters,
V=V,
U0=U0,
rho0=rho0,
t0=t0,
debug=debug,
solver_type = solver_type,
preconditioner_type = preconditioner_type,
periodic=periodic,
ligands=ligands
)
self.debug = debug
self.solver_type = solver_type
self.preconditioner_type = preconditioner_type
self.periodic = False
self.ligands = ligands
self.nligands = ligands.nligands()
self.params = self.default_params.copy()
if (mesh):
self.omesh = self.mesh = mesh
else:
self.omesh = self.mesh = box_mesh(width=width, dim=dim,
nelements=nelements)
self.nelements = nelements
logMULTIPLE('self.mesh', self.mesh)
logMULTIPLE('self.mesh.mpi_comm().size', self.mesh.mpi_comm().size)
self.nelements = nelements
self.degree = degree
self.dim = self.mesh.geometry().dim()
self.params['dim'] = self.dim
self.params.update(parameters)
#
# Solution spaces and Functions
#
fss = self.make_function_space()
(self.SE, self.SS, self.VE, self.VS) = [
fss[fs] for fs in ('SE', 'SS', 'VE', 'VS')
]
logMULTIPLE('self.VS', self.VS)
self.sol = Function(self.VS) # sol, current soln
logMULTIPLE('self.sol', self.sol)
self.srho, self.sUs = self.sol.sub(0), self.sol.split()[1:]
splitsol = fe.split(self.sol)
self.irho, self.iUs = splitsol[0], splitsol[1:]
tfs = TestFunctions(self.VS)
self.wrho, self.wUs = tfs[0], tfs[1:]
self.tdsol = TrialFunction(self.VS)
splittdsol = fe.split(self.tdsol)
self.tdrho, self.tdUs = splittdsol[0], splittdsol[1:]
self.n = FacetNormal(self.mesh)
self.h = CellDiameter(self.mesh)
self.havg = fe.avg(self.h)
self.dx = fe.dx
# self.dx = fe.dx(metadata={'quadrature_degree': min(degree, 10)})
self.dS = fe.dS
# self.dS = fe.dS(metadata={'quadrature_degree': min(degree, 10)})
#
# record initial state
#
try:
V(self.iUs, self.irho)
def realV(Us, rho):
return V(Us, rho)
except TypeError:
def realV(Us, rho):
return V(Us)
self.V = realV
if not U0:
U0 = [Constant(0.0)] * self.nligands
self.U0s = [Constant(0.0)] * self.nligands
for i,U0i in enumerate(U0):
if isinstance(U0i, ufl.coefficient.Coefficient):
self.U0s[i] = U0i
else:
self.U0s[i] = Expression(U0i, **self.params,
degree=self.degree,
domain=self.mesh)
if not rho0:
rho0 = Constant(0.0)
if isinstance(rho0, ufl.coefficient.Coefficient):
self.rho0 = rho0
else:
self.rho0 = Expression(rho0, **self.params,
degree=self.degree, domain=self.mesh)
self.t0 = t0
#
# initialize state
#
logMULTIPLE('restarting')
self.restart()
logMULTIPLE('restart returned')
return(None)
def make_function_space(self,
mesh=None,
dim=None,
degree=None
):
if not mesh: mesh = self.mesh
if not dim: dim = self.dim
if not degree: degree = self.degree
SE = FiniteElement('DG', cellShapes[dim-1], degree)
SS = FunctionSpace(mesh, SE) # scalar space
elements = [SE] * (self.nligands + 1)
VE = MixedElement(elements)
VS = FunctionSpace(mesh, VE) # vector space
return dict(SE=SE, SS=SS, VE=VE, VS=VS)
def restart(self):
logMULTIPLE('restart')
self.t = self.t0
CE = FiniteElement('CG', cellShapes[self.dim-1], self.degree)
CS = FunctionSpace(self.mesh, CE) # scalar space
coords = gather_dof_coords(CS)
fe.assign(self.sol.sub(0),
function_interpolate(self.rho0, self.SS,
coords=coords))
for i,U0i in enumerate(self.U0s):
fe.assign(self.sol.sub(i+1),
function_interpolate(U0i, self.SS, coords=coords))
logMULTIPLE('U0s assign returned')
def setup_problem(self, debug=False):
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
self.drho_integral = self.tdrho*self.wrho*self.dx
self.dU_integral = sum(
[tdUi*wUi*self.dx for tdUi,wUi in zip(self.tdUs, self.wUs)]
)
self.A = fe.assemble(self.drho_integral + self.dU_integral)
self.dsol = Function(self.VS)
dsolsplit = self.dsol.split()
self.drho, self.dUs = dsolsplit[0], dsolsplit[1:]
#
# assemble RHS (for each time point, but compile only once)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.params['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rho_min = self.params['rho_min']
self.rhopen = self.params['rhopen']
self.grhopen = self.params['grhopen']
self.v = -ufl.grad(self.V(self.iUs, self.irho)) - (
self.s2*ufl.grad(self.irho)/ufl.max_value(self.irho,
self.rho_min)
)
self.flux = self.v * self.irho
self.vn = ufl.max_value(ufl.dot(self.v, self.n), 0)
self.facet_flux = (
self.vn('+')*ufl.max_value(self.irho('+'), 0.0) -
self.vn('-')*ufl.max_value(self.irho('-'), 0.0)
)
self.rho_flux_jump = -self.facet_flux*ufl.jump(self.wrho)*self.dS
self.rho_grad_move = ufl.dot(self.flux,
ufl.grad(self.wrho))*self.dx
self.rho_penalty = -(
(self.rhopen * self.degree**2 / self.havg) *
ufl.dot(ufl.jump(self.irho, self.n),
ufl.jump(self.wrho, self.n)) * self.dS
)
self.grho_penalty = -(
self.grhopen * self.degree**2 *
(ufl.jump(ufl.grad(self.irho), self.n) *
ufl.jump(ufl.grad(self.wrho), self.n)) * self.dS
)
self.rho_terms = (
self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty
)
if not hasattr(self, 'U_terms'):
self.U_min = self.params['U_min']
self.Upen = self.params['Upen']
self.gUpen = self.params['gUpen']
self.U_decay = sum(
[-lig.gamma * iUi * wUi * self.dx for
lig,iUi,wUi in
zip(self.ligands.ligands(), self.iUs, self.wUs)]
)
self.U_secretion = sum(
[lig.s * self.irho * wUi * self.dx for
lig,wUi in zip(self.ligands.ligands(), self.wUs)]
)
self.jump_gUw = sum(
[lig.D * ufl.jump(wUi * ufl.grad(iUi), self.n) * self.dS
for lig,wUi,iUi in
zip(self.ligands.ligands(), self.wUs, self.iUs)]
)
self.U_diffusion = sum(
[-lig.D * ufl.dot(ufl.grad(iUi), ufl.grad(wUi))*self.dx for
lig,iUi,wUi in
zip(self.ligands.ligands(), self.iUs, self.wUs)]
)
self.U_penalty = sum(
[-(self.Upen*self.degree**2/self.havg) *
ufl.dot(ufl.jump(iUi, self.n),
ufl.jump(wUi, self.n))*self.dS for
iUi,wUi in zip(self.iUs, self.wUs)]
)
self.gU_penalty = -self.gUpen * self.degree**2 * sum(
[ufl.jump(ufl.grad(iUi), self.n) *
ufl.jump(ufl.grad(wUi), self.n) * self.dS for
iUi,wUi in zip(self.iUs, self.wUs)]
)
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty
)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
def ddt(self, debug=False):
"""Calculate time derivative of rho and U
Results are left in self.dsol as a two-component vector function.
"""
self.setup_problem(debug)
self.b = fe.assemble(self.all_terms)
return fe.solve(self.A, self.dsol.vector(), self.b,
self.solver_type)
def project_gradient(
f0,
mesh=None,
degree=None,
debug=False,
solver_type='gmres',
preconditioner_type='default'
):
"""Find an approximation to f0 that has the same gradient
Parameters:
f0: the function to approximate
mesh=None: the mesh on which to approximate it. If not provided, the
mesh is extracted from f0.
degree=None: degree of the polynomial approximation. extracted
from f0 if not provided.
solver_type='gmres': The linear solver type to use.
preconditioner_type='default': Preconditioner type to use
"""
if not mesh: mesh = f0.function_space().mesh()
element = f0.ufl_element()
if not degree:
degree = element.degree()
CE = FiniteElement('CG', mesh.ufl_cell(), degree)
CS = FunctionSpace(mesh, CE)
DE = FiniteElement('DG', mesh.ufl_cell(), degree)
DS = FunctionSpace(mesh, DE)
CVE = VectorElement('CG', mesh.ufl_cell(), degree - 1)
CV = FunctionSpace(mesh, CVE)
RE = FiniteElement('R', mesh.ufl_cell(), 0)
R = FunctionSpace(mesh, RE)
CRE = MixedElement([CE, RE])
CR = FunctionSpace(mesh, CRE)
f = fe.project(f0, CS,
solver_type=solver_type,
preconditioner_type= preconditioner_type)
g = fe.project(fe.grad(f), CV,
solver_type=solver_type,
preconditioner_type= preconditioner_type)
tf, tc = TrialFunction(CR)
wf, wc = TestFunctions(CR)
dx = Measure('dx', domain=mesh,
metadata={'quadrature_degree': min(degree, 10)})
a = (fe.dot(fe.grad(tf), fe.grad(wf)) + tc * wf + tf * wc) * fe.dx
L = (f * wc + fe.dot(g, fe.grad(wf))) * fe.dx
igc = Function(CR)
fe.solve(a == L, igc,
solver_parameters={'linear_solver': solver_type,
'preconditioner': preconditioner_type}
)
if debug:
print('igc', igc.vector()[:])
assigner = FunctionAssigner(CS, CR.sub(0))
# ig = Function(CS)
# assigner.assign(ig, igc.sub(0))
# fe.assign(ig, igc.sub(0))
# if debug:
# print('ig', igc.sub(0).vector()[:])
igd = fe.project(igc.sub(0), DS,
solver_type=solver_type,
preconditioner_type=preconditioner_type)
if debug:
print('igd', igd.vector()[:])
return igd