def les_smagorinsky_eddy_viscosity():
from firedrake_fluids.les import LES
errors = []
for n in [2, 4, 8, 16, 32]:
mesh = UnitSquareMesh(n, n)
smagorinsky_coefficient = 2.0
filter_width = CellVolume(mesh)**(1.0/2.0) # Square root of element area
fs_exact = FunctionSpace(mesh, "CG", 3)
fs = FunctionSpace(mesh, "CG", 1)
vfs = VectorFunctionSpace(mesh, "CG", 1)
u = Function(vfs).interpolate(Expression(('sin(x[0])', 'cos(x[0])')))
density = Function(fs).interpolate(Expression("1.0"))
exact_solution = project(Expression('pow(%f, 2) * sqrt(2.0*cos(x[0])*cos(x[0]) + 0.5*sin(x[0])*sin(x[0]) + 0.5*sin(x[0])*sin(x[0]))' % smagorinsky_coefficient), fs_exact)
les = LES(mesh, fs, u, density, (smagorinsky_coefficient/filter_width))
# Since the eddy viscosity depends on the filter_width, we need to provide smagorinsky_coefficient/filter_width here
# for the convergence test because we want to compare against the same exact solution throughout.
les.solve()
eddy_viscosity = les.eddy_viscosity
print eddy_viscosity.vector().array()
errors.append(sqrt(assemble(dot(eddy_viscosity - exact_solution, eddy_viscosity - exact_solution) * dx)))
return errors
def les_smagorinsky_eddy_viscosity():
from firedrake_fluids.les import LES
errors = []
for n in [2, 4, 8, 16, 32]:
mesh = UnitSquareMesh(n, n)
smagorinsky_coefficient = 2.0
filter_width = CellVolume(mesh)**(1.0 / 2.0
) # Square root of element area
fs_exact = FunctionSpace(mesh, "CG", 3)
fs = FunctionSpace(mesh, "CG", 1)
vfs = VectorFunctionSpace(mesh, "CG", 1)
u = Function(vfs).interpolate(Expression(('sin(x[0])', 'cos(x[0])')))
density = Function(fs).interpolate(Expression("1.0"))
exact_solution = project(
Expression(
'pow(%f, 2) * sqrt(2.0*cos(x[0])*cos(x[0]) + 0.5*sin(x[0])*sin(x[0]) + 0.5*sin(x[0])*sin(x[0]))'
% smagorinsky_coefficient), fs_exact)
les = LES(mesh, fs, u, density,
(smagorinsky_coefficient / filter_width))
# Since the eddy viscosity depends on the filter_width, we need to provide smagorinsky_coefficient/filter_width here
# for the convergence test because we want to compare against the same exact solution throughout.
les.solve()
eddy_viscosity = les.eddy_viscosity
print eddy_viscosity.vector().array()
errors.append(
sqrt(
assemble(
dot(eddy_viscosity - exact_solution,
eddy_viscosity - exact_solution) * dx)))
return errors
def run(self, array=None, annotate=False, checkpoint=None):
""" Perform the simulation! """
# The solution field defined on the mixed function space
solution = Function(self.W, name="Solution", annotate=annotate)
# The solution from the previous time-step. At t=0, this holds the initial conditions.
solution_old = Function(self.W, name="SolutionOld", annotate=annotate)
# Assign the initial condition
initial_condition = self.get_initial_condition(checkpoint=checkpoint)
solution_old.assign(initial_condition, annotate=annotate)
# Get the test functions
test_functions = TestFunctions(self.W)
w = test_functions[0]; v = test_functions[1]
LOG.info("Test functions created.")
# These are like the TrialFunctions, but are just regular Functions here because we want to solve a non-linear problem
# 'u' and 'h' are the velocity and free surface perturbation, respectively.
functions = split(solution)
u = functions[0]; h = functions[1]
LOG.info("Trial functions created.")
functions_old = split(solution_old)
u_old = functions_old[0]; h_old = functions_old[1]
# Write initial conditions to file
LOG.info("Writing initial conditions to file...")
self.output_functions["Velocity"].assign(solution_old.split()[0], annotate=False)
self.output_files["Velocity"] << self.output_functions["Velocity"]
self.output_functions["FreeSurfacePerturbation"].assign(solution_old.split()[1], annotate=False)
self.output_files["FreeSurfacePerturbation"] << self.output_functions["FreeSurfacePerturbation"]
# Construct the collection of all the individual terms in their weak form.
LOG.info("Constructing form...")
F = 0
theta = self.options["theta"]
dt = self.options["dt"]
dimension = self.options["dimension"]
g_magnitude = self.options["g_magnitude"]
# Is the Velocity field represented by a discontinous function space?
dg = (self.W.sub(0).ufl_element().family() == "Discontinuous Lagrange")
# Mean free surface height
h_mean = Function(self.W.sub(1), name="FreeSurfaceMean", annotate=False)
h_mean.interpolate(ExpressionFromOptions(path = "/system/core_fields/scalar_field::FreeSurfaceMean/value").get_expression())
# Weight u and h by theta to obtain the theta time-stepping scheme.
assert(theta >= 0.0 and theta <= 1.0)
LOG.info("Time-stepping scheme using theta = %g" % (theta))
u_mid = (1.0 - theta) * u_old + theta * u
h_mid = (1.0 - theta) * h_old + theta * h
# The total height of the free surface.
H = h_mean + h
# Simple P1 function space, to be used in the stabilisation routines (if applicable).
P1 = FunctionSpace(self.mesh, "CG", 1)
cellsize = CellSize(self.mesh)
# Normal vector to each element facet
n = FacetNormal(self.mesh)
# Mass term
if(self.options["have_momentum_mass"]):
LOG.debug("Momentum equation: Adding mass term...")
M_momentum = (1.0/dt)*(inner(w, u) - inner(w, u_old))*dx
F += M_momentum
# Advection term
if(self.options["have_momentum_advection"]):
LOG.debug("Momentum equation: Adding advection term...")
if(self.options["integrate_advection_term_by_parts"]):
outflow = (dot(u_mid, n) + abs(dot(u_mid, n)))/2.0
A_momentum = -inner(dot(u_mid, grad(w)), u_mid)*dx - inner(dot(u_mid, grad(u_mid)), w)*dx
A_momentum += inner(w, outflow*u_mid)*ds
if(dg):
# Only add interior facet integrals if we are dealing with a discontinous Galerkin discretisation.
A_momentum += dot(outflow('+')*u_mid('+') - outflow('-')*u_mid('-'), jump(w))*dS
else:
A_momentum = inner(dot(grad(u_mid), u_mid), w)*dx
F += A_momentum
# Viscous stress term. Note that the viscosity is kinematic (not dynamic).
if(self.options["have_momentum_stress"]):
LOG.debug("Momentum equation: Adding stress term...")
viscosity = Function(self.W.sub(1))
# Background viscosity
background_viscosity = Function(self.W.sub(1)).interpolate(Expression(libspud.get_option("/system/equations/momentum_equation/stress_term/scalar_field::Viscosity/value/constant")))
viscosity.assign(background_viscosity)
# Eddy viscosity
if(self.options["have_turbulence_parameterisation"]):
LOG.debug("Momentum equation: Adding turbulence parameterisation...")
base_option_path = "/system/equations/momentum_equation/turbulence_parameterisation"
# Large eddy simulation (LES)
if(libspud.have_option(base_option_path + "/les")):
density = Constant(1.0) # We divide through by density in the momentum equation, so just set this to 1.0 for now.
smagorinsky_coefficient = Constant(libspud.get_option(base_option_path + "/les/smagorinsky/smagorinsky_coefficient"))
les = LES(self.mesh, self.W.sub(1), u_mid, density, smagorinsky_coefficient)
# Add on eddy viscosity
viscosity += les.eddy_viscosity
# Stress tensor: tau = grad(u) + transpose(grad(u)) - (2/3)*div(u)
if(not dg):
# Perform a double dot product of the stress tensor and grad(w).
K_momentum = -viscosity*inner(grad(u_mid) + grad(u_mid).T, grad(w))*dx
K_momentum += viscosity*(2.0/3.0)*inner(div(u_mid)*Identity(dimension), grad(w))*dx
else:
# Interior penalty method
cellsize = Constant(0.2) # In general, we should use CellSize(self.mesh) instead.
alpha = 1/cellsize # Penalty parameter.
K_momentum = -viscosity('+')*inner(grad(u_mid), grad(w))*dx
for dim in range(self.options["dimension"]):
K_momentum += -viscosity('+')*(alpha('+')/cellsize('+'))*dot(jump(w[dim], n), jump(u_mid[dim], n))*dS
K_momentum += viscosity('+')*dot(avg(grad(w[dim])), jump(u_mid[dim], n))*dS + viscosity('+')*dot(jump(w[dim], n), avg(grad(u_mid[dim])))*dS
F -= K_momentum # Negative sign here because we are bringing the stress term over from the RHS.
# The gradient of the height of the free surface, h
LOG.debug("Momentum equation: Adding gradient term...")
C_momentum = -g_magnitude*inner(w, grad(h_mid))*dx
F -= C_momentum
# Quadratic drag term in the momentum equation
if(self.options["have_drag"]):
LOG.debug("Momentum equation: Adding drag term...")
base_option_path = "/system/equations/momentum_equation/drag_term"
# Get the bottom drag/friction coefficient.
LOG.debug("Momentum equation: Adding bottom drag contribution...")
bottom_drag = ExpressionFromOptions(path=base_option_path+"/scalar_field::BottomDragCoefficient/value", t=0).get_expression()
bottom_drag = Function(self.W.sub(1)).interpolate(bottom_drag)
# Magnitude of the velocity field
magnitude = sqrt(dot(u_old, u_old))
# Form the drag term
if(array):
LOG.debug("Momentum equation: Adding turbine drag contribution...")
drag_coefficient = bottom_drag + array.turbine_drag()
else:
drag_coefficient = bottom_drag
D_momentum = -inner(w, (drag_coefficient*magnitude/H)*u_mid)*dx
F -= D_momentum
# The mass term in the shallow water continuity equation
# (i.e. an advection equation for the free surface height, h)
if(self.options["have_continuity_mass"]):
LOG.debug("Continuity equation: Adding mass term...")
M_continuity = (1.0/dt)*(inner(v, h) - inner(v, h_old))*dx
F += M_continuity
# Append any Expression objects for weak BCs here.
weak_bc_expressions = []
# Divergence term in the shallow water continuity equation
LOG.debug("Continuity equation: Adding divergence term...")
if(self.options["integrate_continuity_equation_by_parts"]):
LOG.debug("The divergence term is being integrated by parts.")
Ct_continuity = - H*inner(u_mid, grad(v))*dx
if(dg):
Ct_continuity += inner(jump(v, n), avg(H*u_mid))*dS
# Add in the surface integrals, but check to see if any boundary conditions need to be applied weakly here.
boundary_markers = self.mesh.exterior_facets.unique_markers
for marker in boundary_markers:
marker = int(marker) # ds() will not accept markers of type 'numpy.int32', so convert it to type 'int' here.
bc_type = None
for i in range(0, libspud.option_count("/system/core_fields/vector_field::Velocity/boundary_condition")):
bc_path = "/system/core_fields/vector_field::Velocity/boundary_condition[%d]" % i
if(not (marker in libspud.get_option(bc_path + "/surface_ids"))):
# This BC is not associated with this marker, so skip it.
continue
# Determine the BC type.
if(libspud.have_option(bc_path + "/type::no_normal_flow")):
bc_type = "no_normal_flow"
elif(libspud.have_option(bc_path + "/type::dirichlet")):
if(libspud.have_option(bc_path + "/type::dirichlet/apply_weakly")):
bc_type = "weak_dirichlet"
else:
bc_type = "dirichlet"
elif(libspud.have_option(bc_path + "/type::flather")):
bc_type = "flather"
# Apply the boundary condition...
try:
LOG.debug("Applying Velocity BC of type '%s' to surface ID %d..." % (bc_type, marker))
if(bc_type == "flather"):
# The known exterior value for the Velocity.
u_ext = ExpressionFromOptions(path = (bc_path + "/type::flather/exterior_velocity"), t=0).get_expression()
Ct_continuity += H*inner(Function(self.W.sub(0)).interpolate(u_ext), n)*v*ds(int(marker))
# The known exterior value for the FreeSurfacePerturbation.
h_ext = ExpressionFromOptions(path = (bc_path + "/type::flather/exterior_free_surface_perturbation"), t=0).get_expression()
Ct_continuity += H*sqrt(g_magnitude/H)*(h_mid - Function(self.W.sub(1)).interpolate(h_ext))*v*ds(int(marker))
weak_bc_expressions.append(u_ext)
weak_bc_expressions.append(h_ext)
elif(bc_type == "weak_dirichlet"):
u_bdy = ExpressionFromOptions(path = (bc_path + "/type::dirichlet"), t=0).get_expression()
Ct_continuity += H*(dot(Function(self.W.sub(0)).interpolate(u_bdy), n))*v*ds(int(marker))
weak_bc_expressions.append(u_bdy)
elif(bc_type == "dirichlet"):
# Add in the surface integral as it is here. The BC will be applied strongly later using a DirichletBC object.
Ct_continuity += H * inner(u_mid, n) * v * ds(int(marker))
elif(bc_type == "no_normal_flow"):
# Do nothing here since dot(u, n) is zero.
continue
else:
raise ValueError("Unknown boundary condition type!")
except ValueError as e:
LOG.exception(e)
sys.exit()
# If no boundary condition has been applied, include the surface integral as it is.
if(bc_type is None):
Ct_continuity += H * inner(u_mid, n) * v * ds(int(marker))
else:
Ct_continuity = inner(v, div(H*u_mid))*dx
F += Ct_continuity
# Add in any source terms
if(self.options["have_momentum_source"]):
LOG.debug("Momentum equation: Adding source term...")
momentum_source_expression = ExpressionFromOptions(path = "/system/equations/momentum_equation/source_term/vector_field::Source/value", t=0).get_expression()
momentum_source_function = Function(self.W.sub(0), annotate=False)
F -= inner(w, momentum_source_function.interpolate(momentum_source_expression))*dx
if(self.options["have_continuity_source"]):
LOG.debug("Continuity equation: Adding source term...")
continuity_source_expression = ExpressionFromOptions(path = "/system/equations/continuity_equation/source_term/scalar_field::Source/value", t=0).get_expression()
continuity_source_function = Function(self.W.sub(1), annotate=False)
F -= inner(v, continuity_source_function.interpolate(continuity_source_expression))*dx
# Add in any SU stabilisation
if(self.options["have_su_stabilisation"]):
LOG.debug("Momentum equation: Adding streamline-upwind stabilisation term...")
stabilisation = Stabilisation(self.mesh, P1, cellsize)
magnitude = magnitude_vector(solution_old.split()[0], P1)
# Bound the values for the magnitude below by 1.0e-9 for numerical stability reasons.
u_nodes = magnitude.vector()
near_zero = numpy.array([1.0e-9 for i in range(len(u_nodes))])
u_nodes.set_local(numpy.maximum(u_nodes.array(), near_zero))
diffusivity = ExpressionFromOptions(path = "/system/equations/momentum_equation/stress_term/scalar_field::Viscosity/value", t=self.options["t"]).get_expression()
diffusivity = Function(self.W.sub(1)).interpolate(diffusivity) # Background viscosity
grid_pe = grid_peclet_number(diffusivity, magnitude, P1, cellsize)
# Bound the values for grid_pe below by 1.0e-9 for numerical stability reasons.
grid_pe_nodes = grid_pe.vector()
values = numpy.array([1.0e-9 for i in range(len(grid_pe_nodes))])
grid_pe_nodes.set_local(numpy.maximum(grid_pe_nodes.array(), values))
F += stabilisation.streamline_upwind(w, u, magnitude, grid_pe)
LOG.info("Form construction complete.")
bcs, bc_expressions = self.get_dirichlet_boundary_conditions()
# Prepare solver_parameters dictionary
solver_parameters = self.get_solver_parameters()
# Construct the solver objects
problem = NonlinearVariationalProblem(F, solution, bcs=bcs)
solver = NonlinearVariationalSolver(problem, solver_parameters=solver_parameters)
LOG.debug("Variational problem solver created.")
# PETSc solver run-times
from petsc4py import PETSc
main_solver_stage = PETSc.Log.Stage('Main block-coupled system solve')
total_solver_time = 0.0
# Time-stepping parameters and constants
T = self.options["T"]
t = self.options["t"]
t += dt
iterations_since_dump = 1
iterations_since_checkpoint = 1
# The time-stepping loop
LOG.info("Entering the time-stepping loop...")
#if annotate: adj_start_timestep(time=t)
EPSILON = 1.0e-14
while t <= T + EPSILON: # A small value EPSILON is added here in case of round-off error.
LOG.info("t = %g" % t)
## Update any time-dependent Functions and Expressions.
# Re-compute the velocity magnitude and grid Peclet number fields.
if(self.options["have_su_stabilisation"]):
magnitude.assign(magnitude_vector(solution_old.split()[0], P1))
# Bound the values for the magnitude below by 1.0e-9 for numerical stability reasons.
u_nodes = magnitude.vector()
near_zero = numpy.array([1.0e-9 for i in range(len(u_nodes))])
u_nodes.set_local(numpy.maximum(u_nodes.array(), near_zero))
grid_pe.assign(grid_peclet_number(diffusivity, magnitude, P1, cellsize))
# Bound the values for grid_pe below by 1.0e-9 for numerical stability reasons.
grid_pe_nodes = grid_pe.vector()
values = numpy.array([1.0e-9 for i in range(len(grid_pe_nodes))])
grid_pe_nodes.set_local(numpy.maximum(grid_pe_nodes.array(), values))
if(self.options["have_turbulence_parameterisation"]):
les.solve()
# Time-dependent source terms
if(self.options["have_momentum_source"]):
momentum_source_expression.t = t
momentum_source_function.interpolate(momentum_source_expression)
if(self.options["have_continuity_source"]):
continuity_source_expression.t = t
continuity_source_function.interpolate(continuity_source_expression)
# Update any time-varying DirichletBC objects.
for expr in bc_expressions:
expr.t = t
for expr in weak_bc_expressions:
expr.t = t
# Solve the system of equations!
start_solver_time = mpi4py.MPI.Wtime()
main_solver_stage.push()
LOG.debug("Solving the system of equations...")
solver.solve(annotate=annotate)
main_solver_stage.pop()
end_solver_time = mpi4py.MPI.Wtime()
total_solver_time += (end_solver_time - start_solver_time)
# Write the solution to file.
if((self.options["dump_period"] is not None) and (dt*iterations_since_dump >= self.options["dump_period"])):
LOG.debug("Writing data to file...")
self.output_functions["Velocity"].assign(solution.split()[0], annotate=False)
self.output_files["Velocity"] << self.output_functions["Velocity"]
self.output_functions["FreeSurfacePerturbation"].assign(solution.split()[1], annotate=False)
self.output_files["FreeSurfacePerturbation"] << self.output_functions["FreeSurfacePerturbation"]
iterations_since_dump = 0 # Reset the counter.
# Print out the total power generated by turbines.
if(array):
LOG.info("Power = %.2f" % array.power(u, density=1000))
# Checkpointing
if((self.options["checkpoint_period"] is not None) and (dt*iterations_since_checkpoint >= self.options["checkpoint_period"])):
LOG.debug("Writing checkpoint data to file...")
solution.dat.save("checkpoint")
iterations_since_checkpoint = 0 # Reset the counter.
# Check whether a steady-state has been reached.
if(steady_state(solution.split()[0], solution_old.split()[0], self.options["steady_state_tolerance"]) and steady_state(solution.split()[1], solution_old.split()[1], self.options["steady_state_tolerance"])):
LOG.info("Steady-state attained. Exiting the time-stepping loop...")
break
self.compute_diagnostics()
# Move to next time step
solution_old.assign(solution, annotate=annotate)
#array.turbine_drag.assign(project(array.turbine_drag, array.turbine_drag.function_space(), annotate=annotate), annotate=annotate)
t += dt
#if annotate: adj_inc_timestep(time=t, finished=t>T)
iterations_since_dump += 1
iterations_since_checkpoint += 1
LOG.debug("Moving to next time level...")
LOG.info("Out of the time-stepping loop.")
LOG.debug("Total solver time: %.2f" % (total_solver_time))
return solution
