def sw_solve(config, state, turbine_field=None, functional=None, annotate=True, u_source=None):
'''Solve the shallow water equations with the parameters specified in params.
Options for linear_solver and preconditioner are:
linear_solver: lu, cholesky, cg, gmres, bicgstab, minres, tfqmr, richardson
preconditioner: none, ilu, icc, jacobi, bjacobi, sor, amg, additive_schwarz, hypre_amg, hypre_euclid, hypre_parasails, ml_amg
'''
############################### Setting up the equations ###########################
# Define variables for all used parameters
ds = config.domain.ds
params = config.params
# To begin with, check if the provided parameters are valid
params.check()
theta = params["theta"]
dt = params["dt"]
g = params["g"]
depth = params["depth"]
# Reset the time
params["current_time"] = params["start_time"]
t = params["current_time"]
quadratic_friction = params["quadratic_friction"]
include_advection = params["include_advection"]
include_diffusion = params["include_diffusion"]
include_time_term = params["include_time_term"]
diffusion_coef = params["diffusion_coef"]
newton_solver = params["newton_solver"]
picard_relative_tolerance = params["picard_relative_tolerance"]
picard_iterations = params["picard_iterations"]
linear_solver = params["linear_solver"]
preconditioner = params["preconditioner"]
bctype = params["bctype"]
strong_bc = params["strong_bc"]
free_slip_on_sides = params["free_slip_on_sides"]
steady_state = params["steady_state"]
functional_final_time_only = params["functional_final_time_only"]
functional_quadrature_degree = params["functional_quadrature_degree"]
is_nonlinear = (include_advection or quadratic_friction)
turbine_thrust_parametrisation = params["turbine_thrust_parametrisation"]
implicit_turbine_thrust_parametrisation = params["implicit_turbine_thrust_parametrisation"]
cache_forward_state = params["cache_forward_state"]
if not 0 <= functional_quadrature_degree <= 1:
raise ValueError("functional_quadrature_degree must be 0 or 1.")
if implicit_turbine_thrust_parametrisation:
function_space = config.function_space_2enriched
elif turbine_thrust_parametrisation:
function_space = config.function_space_enriched
else:
function_space = config.function_space
# Take care of the steady state case
if steady_state:
dt = 1.
params["finish_time"] = params["start_time"] + dt / 2
theta = 1.
# Define test functions
if implicit_turbine_thrust_parametrisation:
v, q, o, o_adv = TestFunctions(function_space)
elif turbine_thrust_parametrisation:
v, q, o = TestFunctions(function_space)
else:
v, q = TestFunctions(function_space)
# Define functions
state_new = Function(function_space, name="New_state") # solution of the next timestep
state_nl = Function(function_space, name="Best_guess_state") # the last computed state of the next timestep, used for the picard iteration
if not newton_solver and (turbine_thrust_parametrisation or implicit_turbine_thrust_parametrisation):
raise NotImplementedError("Thrust turbine representation does currently only work with the newton solver.")
# Split mixed functions
if is_nonlinear and newton_solver:
if implicit_turbine_thrust_parametrisation:
u, h, up_u, up_u_adv = split(state_new)
elif turbine_thrust_parametrisation:
u, h, up_u = split(state_new)
else:
u, h = split(state_new)
else:
u, h = TrialFunctions(function_space)
if implicit_turbine_thrust_parametrisation:
u0, h0, up_u0, up_u_adv0 = split(state)
elif turbine_thrust_parametrisation:
u0, h0, up_u0 = split(state)
else:
u0, h0 = split(state)
u_nl, h_nl = split(state_nl)
# Create initial conditions and interpolate
state_new.assign(state, annotate=annotate)
# u_(n+theta) and h_(n+theta)
u_mid = (1.0 - theta) * u0 + theta * u
h_mid = (1.0 - theta) * h0 + theta * h
# If a picard iteration is used we need an intermediate state
if is_nonlinear and not newton_solver:
u_nl, h_nl = split(state_nl)
state_nl.assign(state, annotate=annotate)
u_mid_nl = (1.0 - theta) * u0 + theta * u_nl
# The normal direction
n = FacetNormal(function_space.mesh())
# Mass matrix
M = inner(v, u) * dx
M += inner(q, h) * dx
M0 = inner(v, u0) * dx
M0 += inner(q, h0) * dx
# Divergence term.
Ct_mid = -depth * inner(u_mid, grad(q)) * dx
#+inner(avg(u_mid),jump(q,n))*dS # This term is only needed for dg element pairs
if bctype == 'dirichlet':
if steady_state:
raise ValueError("Can not use a time dependent boundary condition for a steady state simulation")
# The dirichlet boundary condition on the left hand side
expr = config.params["weak_dirichlet_bc_expr"]
bc_contr = - depth * dot(expr, n) * q * ds(1)
# The dirichlet boundary condition on the right hand side
bc_contr -= depth * dot(expr, n) * q * ds(2)
# We enforce a no-normal flow on the sides by removing the surface integral.
# bc_contr -= dot(u_mid, n) * q * ds(3)
elif bctype == 'flather':
if steady_state:
raise ValueError("Can not use a time dependent boundary condition for a steady state simulation")
# The Flather boundary condition on the left hand side
expr = config.params["flather_bc_expr"]
bc_contr = - depth * dot(expr, n) * q * ds(1)
Ct_mid += sqrt(g * depth) * inner(h_mid, q) * ds(1)
# The contributions of the Flather boundary condition on the right hand side
Ct_mid += sqrt(g * depth) * inner(h_mid, q) * ds(2)
elif bctype == 'strong_dirichlet':
# Do not replace anything in the surface integrals as the strong Dirichlet Boundary condition will do that
bc_contr = -depth * dot(u_mid, n) * q * ds(1)
bc_contr -= depth * dot(u_mid, n) * q * ds(2)
if not free_slip_on_sides:
bc_contr -= depth * dot(u_mid, n) * q * ds(3)
else:
info_red("Unknown boundary condition type: %s" % bctype)
sys.exit(1)
# Pressure gradient operator
C_mid = g * inner(v, grad(h_mid)) * dx
#+inner(avg(v),jump(h_mid,n))*dS # This term is only needed for dg element pairs
# Bottom friction
friction = params["friction"]
if turbine_field:
if type(turbine_field) == list:
tf = Function(turbine_field[0], name="turbine_friction", annotate=annotate)
else:
tf = Function(turbine_field, name="turbine_friction", annotate=annotate)
if turbine_thrust_parametrisation or implicit_turbine_thrust_parametrisation:
print0("Adding thrust force")
# Compute the upstream velocities
if implicit_turbine_thrust_parametrisation:
up_u_eq = upstream_u_implicit_equation(config, tf, u, up_u, o, up_u_adv, o_adv)
elif turbine_thrust_parametrisation:
up_u_eq = upstream_u_equation(config, tf, u, up_u, o)
def thrust_force(up_u, min=smooth_uflmin):
''' Returns the thrust force for a given upstream velcocity '''
# Now apply a pointwise transformation based on the interpolation of a loopup table
c_T_coeffs = [0.08344535, -1.42428216, 9.13153605, -26.19370168, 28.8752054]
c_T_coeffs.reverse()
c_T = min(0.88, sum([c_T_coeffs[i] * up_u ** i for i in range(len(c_T_coeffs))]))
# The amount of forcing we want to apply
turbine_radius = 15.
A_c = pi * Constant(turbine_radius ** 2) # Turbine cross section
f = 0.5 * c_T * up_u ** 2 * A_c
return f
# Apply the force in the opposite direction of the flow
f_dir = -thrust_force(up_u) * u / norm_approx(u, alpha=1e-6)
# Distribute this force over the turbine area
thrust = inner(f_dir * tf / (Constant(config.turbine_cache.turbine_integral()) * config.params["depth"]), v) * dx
# Friction term
# With Newton we can simply use a non-linear form
if quadratic_friction and newton_solver:
R_mid = friction / depth * dot(u_mid, u_mid) ** 0.5 * inner(u_mid, v) * dx
if turbine_field and not (turbine_thrust_parametrisation or implicit_turbine_thrust_parametrisation):
R_mid += tf / depth * dot(u_mid, u_mid) ** 0.5 * inner(u_mid, v) * config.site_dx(1)
# With a picard iteration we need to linearise using the best guess
elif quadratic_friction and not newton_solver:
R_mid = friction / depth * dot(u_mid_nl, u_mid_nl) ** 0.5 * inner(u_mid, v) * dx
if turbine_field and not (turbine_thrust_parametrisation or implicit_turbine_thrust_parametrisation):
R_mid += tf / depth * dot(u_mid_nl, u_mid_nl) ** 0.5 * inner(u_mid, v) * config.site_dx(1)
# Use a linear drag
else:
R_mid = friction / depth * inner(u_mid, v) * dx
if turbine_field and not (turbine_thrust_parametrisation or implicit_turbine_thrust_parametrisation):
R_mid += tf / depth * inner(u_mid, v) * config.site_dx(1)
# Advection term
# With a newton solver we can simply use a quadratic form
if include_advection and newton_solver:
Ad_mid = inner(dot(grad(u_mid), u_mid), v) * dx
# With a picard iteration we need to linearise using the best guess
if include_advection and not newton_solver:
Ad_mid = inner(dot(grad(u_mid), u_mid_nl), v) * dx
if include_diffusion:
# Check that we are not using a DG velocity function space, as the facet integrals are not implemented.
if "Discontinuous" in str(function_space.split()[0]):
raise NotImplementedError("The diffusion term for discontinuous elements is not implemented yet.")
D_mid = diffusion_coef * inner(grad(u_mid), grad(v)) * dx
# Create the final form
G_mid = C_mid + Ct_mid + R_mid
# Add the advection term
if include_advection:
G_mid += Ad_mid
# Add the diffusion term
if include_diffusion:
G_mid += D_mid
# Add the source term
if u_source:
G_mid -= inner(u_source, v) * dx
F = dt * G_mid - dt * bc_contr
# Add the time term
if include_time_term and not steady_state:
F += M - M0
if turbine_thrust_parametrisation or implicit_turbine_thrust_parametrisation:
F += up_u_eq
if turbine_field:
F -= thrust
# Preassemble the lhs if possible
use_lu_solver = (linear_solver == "lu")
if not is_nonlinear:
lhs_preass = assemble(dolfin.lhs(F))
# Precompute the LU factorisation
if use_lu_solver:
info("Computing the LU factorisation for later use ...")
if bctype == 'strong_dirichlet':
raise NotImplementedError("Strong boundary condition and reusing LU factorisation is currently not implemented")
lu_solver = LUSolver(lhs_preass)
lu_solver.parameters["reuse_factorization"] = True
solver_parameters = {"linear_solver": linear_solver, "preconditioner": preconditioner}
# Do some parameter checking:
if "dynamic_turbine_friction" in params["controls"]:
if len(config.params["turbine_friction"]) != (params["finish_time"] - t) / dt + 1:
print0("You control the turbine friction dynamically, but your turbine friction parameter is not an array of length 'number of timesteps' (here: %i)." % ((params["finish_time"] - t) / dt + 1))
import sys
sys.exit(1)
############################### Perform the simulation ###########################
if params["dump_period"] > 0:
try:
statewriter_cb = config.statewriter_callback
except AttributeError:
statewriter_cb = None
writer = StateWriter(config, optimisation_iteration=config.optimisation_iteration, callback=statewriter_cb)
if not steady_state and include_time_term:
print0("Writing state to disk...")
writer.write(state)
step = 0
if functional is not None:
if steady_state or functional_final_time_only:
j = 0.
if params["print_individual_turbine_power"]:
j_individual = [0] * len(params["turbine_pos"])
force_individual = [0] * len(params["turbine_pos"])
else:
if functional_quadrature_degree == 0:
quad = 0.0
else:
quad = 0.5
j = dt * quad * assemble(functional.Jt(state, tf))
if params["print_individual_turbine_power"]:
j_individual = []
force_individual = []
for i in range(len(params["turbine_pos"])):
j_individual.append(dt * quad * assemble(functional.Jt_individual(state, i)))
force_individual.append(dt * quad * assemble(functional.force_individual(state, i)))
print0("Start of time loop")
adjointer.time.start(t)
timestep = 0
while (t < params["finish_time"]):
timestep += 1
t += dt
params["current_time"] = t
# Update bc's
if bctype == "strong_dirichlet":
strong_bc.update_time(t)
else:
expr.t = t - (1.0 - theta) * dt
# Update source term
if u_source:
u_source.t = t - (1.0 - theta) * dt
step += 1
# Solve non-linear system with a Newton sovler
if is_nonlinear and newton_solver:
# Use a Newton solver to solve the nonlinear problem.
#solver_parameters["linear_solver"] = "gmres"
#solver_parameters["linear_solver"] = "superlu_dist"
#solver_parameters["preconditioner"] = "ilu" # does not work in parallel
#solver_parameters["preconditioner"] = "amg"
#solver_parameters["linear_solver"] = "mumps"
#solver_parameters["linear_solver"] = "umfpack"
solver_parameters["newton_solver"] = {}
solver_parameters["newton_solver"]["error_on_nonconvergence"] = True
solver_parameters["newton_solver"]["maximum_iterations"] = 20
solver_parameters["newton_solver"]["convergence_criterion"] = "incremental"
solver_parameters["newton_solver"]["relative_tolerance"] = 1e-16
if cache_forward_state and state_cache.has_key(t):
print0("Load initial guess from cache for time %f." % t)
# Load initial guess for solver from cache
state_new.assign(state_cache[t], annotate=False)
elif not include_time_term:
print0("Set the initial guess for the nonlinear solver to the initial condition.")
# Reset the initial guess after each timestep
ic = config.params['initial_condition']
state_new.assign(ic, annotate=False)
info_blue("Solve shallow water equations at time %s (Newton iteration) ..." % params["current_time"])
if bctype == 'strong_dirichlet':
solve(F == 0, state_new, bcs=strong_bc.bcs, solver_parameters=solver_parameters, annotate=annotate)
else:
solve(F == 0, state_new, solver_parameters=solver_parameters, annotate=annotate)
if turbine_thrust_parametrisation or implicit_turbine_thrust_parametrisation:
print0("Inflow velocity: ", u[0]((10, 160)))
print0("Estimated upstream velocity: ", up_u((640. / 3, 160)))
print0("Expected thrust force: ", thrust_force(u[0]((10, 160)), min=min)((0)))
print0("Total amount of thurst force applied: ", assemble(inner(Constant(1), thrust_force(up_u) * tf / config.turbine_cache.turbine_integral()) * dx))
us.append(u[0]((10, 160)))
thrusts.append(thrust_force(u[0]((10, 160)))((0)))
thrusts_est.append(assemble(inner(Constant(1), thrust_force(up_u) * tf / config.turbine_cache.turbine_integral()) * dx))
import matplotlib.pyplot as plt
plt.clf()
plt.plot(us, thrusts, label="Analytical")
plt.plot(us, thrusts_est, label="Approximated")
plt.legend(loc=2)
plt.savefig("thrust_plot.pdf", format='pdf')
# Solve non-linear system with a Picard iteration
elif is_nonlinear:
# Solve the problem using a picard iteration
iter_counter = 0
while True:
info_blue("Solving shallow water equations at time %s (Picard iteration %d) ..." % (params["current_time"], iter_counter))
if bctype == 'strong_dirichlet':
solve(dolfin.lhs(F) == dolfin.rhs(F), state_new, bcs=strong_bc.bcs, solver_parameters=solver_parameters)
else:
solve(dolfin.lhs(F) == dolfin.rhs(F), state_new, solver_parameters=solver_parameters, annotate=annotate)
iter_counter += 1
if iter_counter > 0:
relative_diff = abs(assemble(inner(state_new - state_nl, state_new - state_nl) * dx)) / assemble(inner(state_new, state_new) * dx)
info_blue("Picard iteration " + str(iter_counter) + " relative difference: " + str(relative_diff))
if relative_diff < picard_relative_tolerance:
info("Picard iteration converged after " + str(iter_counter) + " iterations.")
break
elif iter_counter >= picard_iterations:
info_red("Picard iteration reached maximum number of iterations (" + str(picard_iterations) + ") with a relative difference of " + str(relative_diff) + ".")
break
state_nl.assign(state_new)
# Solve linear system with preassembled matrices
else:
# dolfin can't assemble empty forms which can sometimes happen here.
# A simple workaround is to add a dummy term:
dummy_term = Constant(0) * q * dx
rhs_preass = assemble(dolfin.rhs(F + dummy_term))
# Apply dirichlet boundary conditions
info_blue("Solving shallow water equations at time %s (preassembled matrices) ..." % (params["current_time"]))
if bctype == 'strong_dirichlet':
[bc.apply(lhs_preass, rhs_preass) for bc in strong_bc.bcs]
if use_lu_solver:
info("Using a LU solver to solve the linear system.")
lu_solver.solve(state.vector(), rhs_preass, annotate=annotate)
else:
solve(lhs_preass, state_new.vector(), rhs_preass, solver_parameters["linear_solver"], solver_parameters["preconditioner"], annotate=annotate)
# After the timestep solve, update state
state.assign(state_new)
if cache_forward_state:
# Save state for initial guess cache
print0("Cache initial guess for time %f." % t)
if not state_cache.has_key(t):
state_cache[t] = Function(state_new.function_space())
state_cache[t].assign(state_new, annotate=False)
# Set the control function for the upcoming timestep.
if turbine_field:
if type(turbine_field) == list:
tf.assign(turbine_field[timestep])
else:
tf.assign(turbine_field)
if params["dump_period"] > 0 and step % params["dump_period"] == 0:
print0("Write state to disk...")
writer.write(state)
if functional is not None:
if not (functional_final_time_only and t < params["finish_time"]):
if steady_state or functional_final_time_only or functional_quadrature_degree == 0:
quad = 1.0
elif t >= params["finish_time"]:
quad = 0.5 * dt
else:
quad = 1.0 * dt
j += quad * assemble(functional.Jt(state, tf))
if params["print_individual_turbine_power"]:
info_green("Computing individual turbine power extraction contribution...")
individual_contribution_list = ['x_pos', 'y_pos', 'turbine_power', 'total_force_on_turbine', 'turbine_friction']
fr_individual = range(len(params["turbine_pos"]))
for i in range(len(params["turbine_pos"])):
j_individual[i] += dt * quad * assemble(functional.Jt_individual(state, i))
force_individual[i] += dt * quad * assemble(functional.force_individual(state, i))
if len(params["turbine_friction"]) > 0:
fr_individual[i] = params["turbine_friction"][i]
else:
fr_individual = [params["turbine_friction"]] * len(params["turbine_pos"])
individual_contribution_list.append((params["turbine_pos"][i])[0])
individual_contribution_list.append((params["turbine_pos"][i])[1])
individual_contribution_list.append(j_individual[i])
individual_contribution_list.append(force_individual[i])
individual_contribution_list.append(fr_individual[i])
print0("Contribution of turbine number %d at co-ordinates:" % (i + 1), params["turbine_pos"][i], ' is: ', j_individual[i] * 0.001, 'kW', 'with friction of', fr_individual[i])
# Increase the adjoint timestep
adj_inc_timestep(time=t, finished=(not t < params["finish_time"]))
print0("End of time loop.")
# Write the turbine positions, power extraction and friction to a .csv file named turbine_info.csv
if params['print_individual_turbine_power']:
f = config.params['base_path'] + os.path.sep + "iter_" + str(config.optimisation_iteration) + '/'
# Save the very first result in a different file
if config.optimisation_iteration == 0 and not os.path.isfile(f):
f += 'initial_turbine_info.csv'
else:
f += 'turbine_info.csv'
output_turbines = open(f, 'w')
for i in range(0, len(individual_contribution_list), 5):
print >> output_turbines, '%s, %s, %s, %s, %s' % (individual_contribution_list[i], individual_contribution_list[i + 1], individual_contribution_list[i + 2], individual_contribution_list[i + 3], individual_contribution_list[i + 4])
print 'Total of individual turbines is', sum(j_individual)
if functional is not None:
return j
def solve(self):
# Fetch the function spaces
V, Q = self.problem.parameters.discretisation
# Set up the velocity functions
u = TrialFunction(V)
v = TestFunction(V)
u0 = Function(V)
u1 = Function(V)
# Set up the pressure functions
p = TrialFunction(Q)
q = TestFunction(Q)
p1 = Function(Q)
# Define coefficients
k = Constant(self.problem.parameters.dt)
f = Constant((0,0))
nu = Constant(self.problem.parameters.viscosity)
# Tentative velocity step
F1 = (1/k) * inner(u-u0, v)*dx \
+ inner(grad(u0)*u0, v)*dx \
+ nu * inner(grad(u), grad(v))*dx \
- inner(f, v)*dx
a1 = lhs(F1)
L1 = rhs(F1)
A1 = assemble(a1)
# Pressure update
a2 = inner(grad(p), grad(q))*dx
L2 = (-1./k) * div(u1) * q * dx
A2 = assemble(a2)
# Velocity update
a3 = inner(u, v)*dx
L3 = inner(u1, v)*dx - k * inner(grad(p1), v)*dx
A3 = assemble(a3)
# Iterate over timesteps, start at dt
self.t = self.problem.parameters.dt
while self.t < self.problem.parameters.finish_time:
print 'Starting timestep at time = ', self.t
# Update the boundary conditions for the current timestep
self.bcs.update_time(t=self.t)
# Tentative velocity step
print 'Step 1: Computing tentative velocity'
b1 = assemble(L1)
[bc.apply(A1, b1) for bc in self.bcs.bc_u]
solve(A1, u1.vector(), b1, "gmres", "default")
end()
# Pressure update
print 'Step 2: Updating pressure'
b2 = assemble(L2)
[bc.apply(A2, b2) for bc in self.bcs.bc_p]
solve(A2, p1.vector(), b2, "gmres", self.parameters.prec)
end()
# Velocity update
print 'Step 3: Updating velocity'
b3 = assemble(L3)
[bc.apply(A3, b3) for bc in self.bcs.bc_u]
solve(A3, u1.vector(), b3, "gmres", "default")
end()
# Timestep update
print 'Step 4: Dump and update timestep'
if (self.t/self.problem.parameters.dt)%self.parameters.dump_period == 0:
writer = StateWriter(solver=self)
writer.write(u1, p1)
u0.assign(u1)
self.t += self.problem.parameters.dt
if self.parameters.live_plotting:
plot(p1, title="Pressure", rescale=True)
plot(u1, title="Velocity", rescale=True)
if self.parameters.live_plotting:
interactive()
def solve(self):
# Fetch the function spaces
V, Q = self.problem.parameters.discretisation
# Set up the velocity functions
u = TrialFunction(V)
v = TestFunction(V)
u0 = Function(V)
u1 = Function(V)
# Set up the pressure functions
p = TrialFunction(Q)
q = TestFunction(Q)
p1 = Function(Q)
# Define coefficients
k = Constant(self.problem.parameters.dt)
f = Constant((0, 0))
nu = Constant(self.problem.parameters.viscosity)
# Tentative velocity step
F1 = (1/k) * inner(u-u0, v)*dx \
+ inner(grad(u0)*u0, v)*dx \
+ nu * inner(grad(u), grad(v))*dx \
- inner(f, v)*dx
a1 = lhs(F1)
L1 = rhs(F1)
A1 = assemble(a1)
# Pressure update
a2 = inner(grad(p), grad(q)) * dx
L2 = (-1. / k) * div(u1) * q * dx
A2 = assemble(a2)
# Velocity update
a3 = inner(u, v) * dx
L3 = inner(u1, v) * dx - k * inner(grad(p1), v) * dx
A3 = assemble(a3)
# Iterate over timesteps, start at dt
self.t = self.problem.parameters.dt
while self.t < self.problem.parameters.finish_time:
print 'Starting timestep at time = ', self.t
# Update the boundary conditions for the current timestep
self.bcs.update_time(t=self.t)
# Tentative velocity step
print 'Step 1: Computing tentative velocity'
b1 = assemble(L1)
[bc.apply(A1, b1) for bc in self.bcs.bc_u]
solve(A1, u1.vector(), b1, "gmres", "default")
end()
# Pressure update
print 'Step 2: Updating pressure'
b2 = assemble(L2)
[bc.apply(A2, b2) for bc in self.bcs.bc_p]
solve(A2, p1.vector(), b2, "gmres", self.parameters.prec)
end()
# Velocity update
print 'Step 3: Updating velocity'
b3 = assemble(L3)
[bc.apply(A3, b3) for bc in self.bcs.bc_u]
solve(A3, u1.vector(), b3, "gmres", "default")
end()
# Timestep update
print 'Step 4: Dump and update timestep'
if (self.t / self.problem.parameters.dt
) % self.parameters.dump_period == 0:
writer = StateWriter(solver=self)
writer.write(u1, p1)
u0.assign(u1)
self.t += self.problem.parameters.dt
if self.parameters.live_plotting:
plot(p1, title="Pressure", rescale=True)
plot(u1, title="Velocity", rescale=True)
if self.parameters.live_plotting:
interactive()
