def __init__(self, optimization_algorithm):
"""Initializes the line search
Parameters
----------
optimization_algorithm : cashocs._shape_optimization.shape_optimization_algorithm.ShapeOptimizationAlgorithm
the optimization problem of interest
"""
self.optimization_algorithm = optimization_algorithm
self.config = self.optimization_algorithm.config
self.optimization_problem = self.optimization_algorithm.optimization_problem
self.form_handler = self.optimization_problem.form_handler
self.mesh_handler = self.optimization_problem.mesh_handler
self.deformation = fenics.Function(self.form_handler.deformation_space)
self.stepsize = self.config.getfloat('OptimizationRoutine',
'initial_stepsize',
fallback=1.0)
self.epsilon_armijo = self.config.getfloat('OptimizationRoutine',
'epsilon_armijo',
fallback=1e-4)
self.beta_armijo = self.config.getfloat('OptimizationRoutine',
'beta_armijo',
fallback=2.0)
self.armijo_stepsize_initial = self.stepsize
self.cost_functional = self.optimization_problem.reduced_cost_functional
self.gradient = self.optimization_algorithm.gradient
self.algorithm = _optimization_algorithm_configuration(self.config)
self.is_newton_like = (self.algorithm == 'lbfgs')
self.is_newton = self.algorithm in ['newton']
self.is_steepest_descent = (self.algorithm == 'gradient_descent')
if self.is_newton:
self.stepsize = 1.0
def solve_pde(self, name='pde'):
u = fe.Function(self.function_space)
v = fe.TestFunction(self.function_space)
u_old = fe.project(self.u_0, self.function_space)
# Todo: Average right hand side if we choose it non-constant
flux = self.dt * fe.dot(
self.composed_diff_coef *
(fe.grad(u) + self.drift_function(u)), fe.grad(v)) * fe.dx
bilin_part = u * v * fe.dx + flux
funtional_part = self.rhs * v * self.dt * fe.dx + u_old * v * fe.dx
full_form = bilin_part - funtional_part
num_steps = int(self.T / self.dt) + 1
bc = fe.DirichletBC(self.function_space, self.u_boundary,
MembraneSimulator.full_boundary)
for n in range(num_steps):
print("Step %d" % n)
self.time += self.dt
fe.solve(full_form == 0, u, bc)
fe.plot(u)
plt.savefig('images/plt%d.pdf' % n)
plt.show()
# print(fe.errornorm(u_old, u))
u_old.assign(u)
self.file << (u, self.time)
plt.figure()
# u=u+1E-9
f = fe.plot(u)
# f = fe.plot(u,norm=colors.LogNorm(vmin=1E-9,vmax=2E-4))
plt.rc('text', usetex=True)
plt.colorbar(f, format='%.0e')
plt.title(r'Macroscopic density $u(x,t=1)$ for $\alpha=%.1f$' %
MembraneSimulator.alpha)
plt.xlabel(r'$x_1$')
plt.ylabel(r'$x_2$')
plt.savefig('%s.pdf' % name)
plt.show()
def __call__(self, u_test, return_sol=False, return_permeability=False):
# Pad with zeros
u_test = np.pad(u_test, (0, len(self.indices) - len(u_test)))
# Assembling diffusivity
log_coeff = sum([ui * fi for ui, fi in zip(u_test, self.functions)], 0)
if return_permeability:
return log_coeff
ccode_coeff = sym.ccode(sym.exp(log_coeff))
diff = fen.Expression(ccode_coeff, degree=2)
# Define bilinear form in variational problem
a_bil = diff * fen.dot(fen.grad(self.trial_f), fen.grad(
self.test_f)) * fen.dx
# Compute solution
sol = fen.Function(self.f_space)
fen.solve(a_bil == self.lin_func, sol, self.bound_cond)
evaluations = [sol(xi, yi) for xi, yi in zip(self.x_obs, self.y_obs)]
# print("Evaluating G...:", np.array(evaluations))
return sol if return_sol else np.array(evaluations)
def __call__(self, mesh, V=None, Vc=None):
boundaries, boundarymarkers, domainmarkers, dx, ds, V, Vc = SetupUnitMeshHelper(
mesh, V, Vc)
self._ds_markers = boundarymarkers
self._dx_markers = domainmarkers
self._dx = dx
self._ds = ds
self._boundaries = boundaries
self._subdomains = self._boundaries
u = df.TrialFunction(V)
v = df.TestFunction(V)
alpha = df.Function(Vc, name='conductivity')
alpha.vector()[:] = 1
def set_alpha(x):
alpha.vector()[:] = x
self._set_alpha = set_alpha
a = alpha * df.inner(df.grad(u), df.grad(v)) * dx
return a, set_alpha, alpha, V, Vc
def __init__(self,
mesh: fe.Mesh,
density: fe.Expression,
constitutive_model: ConstitutiveModelBase,
bf: fe.Expression = fe.Expression('0', degree=0)):
super().__init__(mesh, density, constitutive_model, bf)
W = fe.VectorFunctionSpace(mesh, "P", 1)
# Unknowns, values at previous step and test functions
self.w = fe.Function(W)
self.u, self.u0 = self.w, fe.Function(W)
self.v, self.v0 = fe.Function(W), fe.Function(W)
self.a, self.a0 = fe.Function(W), fe.Function(W)
# self.a0 = fe.Function(fe.FunctionSpace(mesh, element_v))
self.ut = fe.TestFunction(W)
self.F = kin.def_grad(self.u)
self.F0 = kin.def_grad(self.u0)
def __init__(self,
mesh: fe.Mesh,
constitutive_model: ConstitutiveModelBase,
density: fe.Expression = fe.Expression('0', degree=0),
bf: fe.Expression = fe.Expression('0', degree=0),
user_output_fn: callable = None):
W = fe.VectorFunctionSpace(mesh, "P", 1)
super().__init__(mesh, constitutive_model, W, bf)
self._density = density
self.user_output_fn = user_output_fn
# Unknowns, values at previous step and test functions
self.w = fe.Function(W)
self.u, self.u0 = self.w, fe.Function(W)
self.v, self.v0 = fe.Function(W), fe.Function(W)
self.a, self.a0 = fe.Function(W), fe.Function(W)
# self.a0 = fe.Function(fe.FunctionSpace(mesh, element_v))
self.ut = fe.TestFunction(W)
self.F = kin.def_grad(self.u)
self.F0 = kin.def_grad(self.u0)
self.output_fn_map = {
Outputs.stress: self.write_stress,
Outputs.strain: self.write_strain,
Outputs.displacement: self.write_u
}
self.d_LHS = fe.inner(self.F * constitutive_model.stress(self.u),
fe.grad(self.ut)) * fe.dx
self.d_RHS = (fe.inner(fe.Constant((0., 0., 0.)), self.ut) * fe.dx)
def runModelButt(self):
if len(markers) > 0:
try:
self.runButt.setEnabled(False)
self.dr = float(self.sptlResLineEdit.text()) # dr = 150
self.pauseButt.setEnabled(True)
interpolateData(True, self.dr,{})
###########################################
### INTERPOLATE DATA SO EVEN INTERVAL ###
###########################################
'''
Interpolating data at an even interval so the data points align with the
invterval mesh.
'''
self.thickness1dInterp = interp1d(thickness.distanceData, thickness.pathData)
self.bed1dInterp = interp1d(bed.distanceData, bed.pathData)
self.surface1dInterp = interp1d(surface.distanceData, surface.pathData)
self.smb1dInterp = interp1d(smb.distanceData, smb.pathData)
self.velocity1dInterp = interp1d(velocity.distanceData, velocity.pathData)
self.t2m1dInterp = interp1d(t2m.distanceData, t2m.pathData)
# N is the number of total data points including the last
# Data points on interval [0, N*dr] inclusive on both ends
self.N = int(np.floor(bed.distanceData[-1] / float(self.dr))) # length of path / resolution
self.x = np.arange(0, (self.N + 1) * self.dr, self.dr) # start point, end point, number of segments. END POINT NOT INCLUDED!
self.mesh = fc.IntervalMesh(self.N, 0, self.dr * self.N) # number of cells, start point, end point
self.thicknessModelData = self.thickness1dInterp(self.x)
self.bedModelData = self.bed1dInterp(self.x)
self.surfaceModelData = self.surface1dInterp(self.x)
self.smbModelData = self.smb1dInterp(self.x)
self.velocityModelData = self.velocity1dInterp(self.x)
self.t2mModelData = self.t2m1dInterp(self.x)
self.THICKLIMIT = 10. # Ice is never less than this thick
self.H = self.surfaceModelData - self.bedModelData
self.surfaceModelData[self.H <= self.THICKLIMIT] = self.bedModelData[self.H <= self.THICKLIMIT]
# FIXME the intervalMesh is consistantly 150 between each datapoint this not true for the data being sent
self.hdf_name = '.data/latest_profile.h5'
self.hfile = fc.HDF5File(self.mesh.mpi_comm(), self.hdf_name, "w")
self.V = fc.FunctionSpace(self.mesh, "CG", 1)
self.functThickness = fc.Function(self.V, name="Thickness")
self.functBed = fc.Function(self.V, name="Bed")
self.functSurface = fc.Function(self.V, name="Surface")
self.functSMB = fc.Function(self.V, name='SMB')
self.functVelocity = fc.Function(self.V, name='Velocity')
self.functT2m = fc.Function(self.V, name='T2m')
surface.pathPlotItem.setData(self.x, self.surfaceModelData)
pg.QtGui.QApplication.processEvents()
self.functThickness.vector()[:] = self.thicknessModelData
self.functBed.vector()[:] = self.bedModelData
self.functSurface.vector()[:] = self.surfaceModelData
self.functSMB.vector()[:] = self.smbModelData
self.functVelocity.vector()[:] = self.velocityModelData
self.functT2m.vector()[:] = self.t2mModelData
self.hfile.write(self.functThickness.vector(), "/thickness")
self.hfile.write(self.functBed.vector(), "/bed")
self.hfile.write(self.functSurface.vector(), "/surface")
self.hfile.write(self.functSMB.vector(), "/smb")
self.hfile.write(self.functVelocity.vector(), "/velocity")
self.hfile.write(self.functT2m.vector(), "/t2m")
self.hfile.write(self.mesh, "/mesh")
self.hfile.close()
self.runModel()
except ValueError:
print 'ERROR: Must have valid spatial resolution.'
def boundary(x, on_boundary):
return on_boundary
bc = fex.DirichletBC(V, u_D, boundary)
# Define variational problem
u = fex.TrialFunction(V)
v = fex.TestFunction(V)
f = fex.Constant(-6.0)
a = fex.dot(fex.grad(u), fex.grad(v)) * fex.dx
L = f * v * fex.dx
# Compute solution
u = fex.Function(V)
fex.solve(a == L, u, bc)
print(u)
# Plot solution and mesh
plt.subplot(2, 1, 1)
fex.plot(u)
#define the boundary conditions
plt.subplot(2, 1, 2)
fex.plot(mesh)
plt.savefig('possion_fex2.png')
#define the boundary conditions
# Save solution to file in VTK format
vtkfile = fex.File('poisson/solution.pvd')
vtkfile << u
# Compute error in L2 norm
def staggered_solve(self):
self.U = fe.VectorFunctionSpace(self.mesh, 'CG', 1)
self.W = fe.FunctionSpace(self.mesh, 'CG', 1)
self.WW = fe.FunctionSpace(self.mesh, 'DG', 0)
self.EE = fe.TensorFunctionSpace(self.mesh, 'DG', 0)
self.MM = fe.VectorFunctionSpace(self.mesh, 'CG', 1)
self.eta = fe.TestFunction(self.U)
self.zeta = fe.TestFunction(self.W)
q = fe.TestFunction(self.WW)
del_x = fe.TrialFunction(self.U)
del_d = fe.TrialFunction(self.W)
p = fe.TrialFunction(self.WW)
self.x_new = fe.Function(self.U, name="u")
self.d_new = fe.Function(self.W, name="d")
self.d_pre = fe.Function(self.W)
self.x_pre = fe.Function(self.U)
x_old = fe.Function(self.U)
d_old = fe.Function(self.W)
self.H_old = fe.Function(self.WW)
self.map_plot = fe.Function(self.MM, name="m")
e = fe.Function(self.EE, name="e")
self.create_custom_xdmf_files()
self.file_results = fe.XDMFFile('data/xdmf/{}/u.xdmf'.format(
self.case_name))
self.file_results.parameters["functions_share_mesh"] = True
vtkfile_e = fe.File('data/pvd/simulation/{}/e.pvd'.format(
self.case_name))
vtkfile_u = fe.File('data/pvd/simulation/{}/u.pvd'.format(
self.case_name))
vtkfile_d = fe.File('data/pvd/simulation/{}/d.pvd'.format(
self.case_name))
for i, (disp, rp) in enumerate(
zip(self.displacements, self.relaxation_parameters)):
print('\n')
print(
'================================================================================='
)
print('>> Step {}, disp boundary condition = {} [mm]'.format(
i, disp))
print(
'================================================================================='
)
self.i = i
self.update_weak_form_due_to_Model_C_bug()
if self.update_weak_form:
self.set_bcs_staggered()
print("Update weak form...")
self.build_weak_form_staggered()
print("Taking derivatives of weak form...")
J_u = fe.derivative(self.G_u, self.x_new, del_x)
J_d = fe.derivative(self.G_d, self.d_new, del_d)
print("Define nonlinear problems...")
p_u = fe.NonlinearVariationalProblem(self.G_u, self.x_new,
self.BC_u, J_u)
p_d = fe.NonlinearVariationalProblem(self.G_d, self.d_new,
self.BC_d, J_d)
print("Define solvers...")
solver_u = fe.NonlinearVariationalSolver(p_u)
solver_d = fe.NonlinearVariationalSolver(p_d)
self.update_weak_form = False
print("Update history weak form")
a = p * q * fe.dx
L = history(self.H_old, self.update_history(),
self.psi_cr) * q * fe.dx
if self.map_flag:
self.interpolate_map()
# delta_x = self.x - self.x_hat
# self.map_plot.assign(fe.project(delta_x, self.MM))
self.presLoad.t = disp
newton_prm = solver_u.parameters['newton_solver']
newton_prm['maximum_iterations'] = 100
# newton_prm['absolute_tolerance'] = 1e-8
newton_prm['relaxation_parameter'] = rp
newton_prm = solver_d.parameters['newton_solver']
newton_prm['maximum_iterations'] = 100
# newton_prm['absolute_tolerance'] = 1e-8
newton_prm['relaxation_parameter'] = rp
vtkfile_e_staggered = fe.File(
'data/pvd/simulation/{}/step{}/e.pvd'.format(
self.case_name, i))
vtkfile_u_staggered = fe.File(
'data/pvd/simulation/{}/step{}/u.pvd'.format(
self.case_name, i))
vtkfile_d_staggered = fe.File(
'data/pvd/simulation/{}/step{}/d.pvd'.format(
self.case_name, i))
iteration = 0
err = 1.
while err > self.staggered_tol:
iteration += 1
solver_d.solve()
solver_u.solve()
if self.solution_scheme == 'explicit':
break
# # Remarks(Tianju): self.x_new.vector() does not behave as expected: producing nan values
# The following lines of codes cause issues
# We use an error measure similar in https://doi.org/10.1007/s10704-019-00372-y
# np_x_new = np.asarray(self.x_new.vector())
# np_d_new = np.asarray(self.d_new.vector())
# np_x_old = np.asarray(x_old.vector())
# np_d_old = np.asarray(d_old.vector())
# err_x = np.linalg.norm(np_x_new - np_x_old) / np.sqrt(len(np_x_new))
# err_d = np.linalg.norm(np_d_new - np_d_old) / np.sqrt(len(np_d_new))
# err = max(err_x, err_d)
# # Remarks(Tianju): dolfin (2019.1.0) errornorm function has severe bugs not behave as expected
# The bug seems to be fixed in later versions
# The following sometimes produces nonzero results in dolfin (2019.1.0)
# print(fe.errornorm(self.d_new, self.d_new, norm_type='l2'))
err_x = fe.errornorm(self.x_new, x_old, norm_type='l2')
err_d = fe.errornorm(self.d_new, d_old, norm_type='l2')
err = max(err_x, err_d)
x_old.assign(self.x_new)
d_old.assign(self.d_new)
e.assign(
fe.project(strain(self.mfem_grad(self.x_new)), self.EE))
print(
'---------------------------------------------------------------------------------'
)
print(
'>> iteration. {}, err_u = {:.5}, err_d = {:.5}, error = {:.5}'
.format(iteration, err_x, err_d, err))
print(
'---------------------------------------------------------------------------------'
)
# vtkfile_e_staggered << e
# vtkfile_u_staggered << self.x_new
# vtkfile_d_staggered << self.d_new
if err < self.staggered_tol or iteration >= self.staggered_maxiter:
print(
'================================================================================='
)
print('\n')
break
print("L2 projection to update the history function...")
fe.solve(a == L, self.H_old, [])
# self.d_pre.assign(self.d_new)
# self.H_old.assign(fe.project(history(self.H_old, self.update_history(), self.psi_cr), self.WW))
if self.map_flag and not self.finish_flag:
self.update_map()
if self.compute_and_save_intermediate_results:
print("Save files...")
self.file_results.write(e, i)
self.file_results.write(self.x_new, i)
self.file_results.write(self.d_new, i)
self.file_results.write(self.map_plot, i)
vtkfile_e << e
vtkfile_u << self.x_new
vtkfile_d << self.d_new
# Assume boundary is not affected by the map.
# There's no need to use the mfem_grad wrapper so that fe.grad is used for speed-up
print("Define forces...")
sigma = cauchy_stress_plus(strain(fe.grad(self.x_new)),
self.psi)
sigma_minus = cauchy_stress_minus(strain(fe.grad(self.x_new)),
self.psi_minus)
sigma_plus = cauchy_stress_plus(strain(fe.grad(self.x_new)),
self.psi_plus)
sigma_degraded = g_d(self.d_new) * sigma_plus + sigma_minus
print("Compute forces...")
if self.case_name == 'pure_shear':
f_full = float(fe.assemble(sigma[0, 1] * self.ds(1)))
f_degraded = float(
fe.assemble(sigma_degraded[0, 1] * self.ds(1)))
else:
f_full = float(fe.assemble(sigma[1, 1] * self.ds(1)))
f_degraded = float(
fe.assemble(sigma_degraded[1, 1] * self.ds(1)))
print("Force full is {}".format(f_full))
print("Force degraded is {}".format(f_degraded))
self.delta_u_recorded.append(disp)
self.force_full.append(f_full)
self.force_degraded.append(f_degraded)
# if force_upper < 0.5 and i > 10:
# break
if self.display_intermediate_results and i % 10 == 0:
self.show_force_displacement()
self.save_data_in_loop()
if self.display_intermediate_results:
plt.ioff()
plt.show()
def demo64(self, permeability, obs_case=1):
mesh = UnitSquareMesh(63, 63)
ak_values = permeability
flux_order = 3
s = scenarios.darcy_problem_1()
DRT = fenics.FiniteElement("DRT", mesh.ufl_cell(), flux_order)
# Lagrange
CG = fenics.FiniteElement("CG", mesh.ufl_cell(), flux_order + 1)
if s.integral_constraint:
# From https://fenicsproject.org/qa/14184/how-to-solve-linear-system-with-constaint
R = fenics.FiniteElement("R", mesh.ufl_cell(), 0)
W = fenics.FunctionSpace(mesh, fenics.MixedElement([DRT, CG, R]))
# Define trial and test functions
(sigma, u, r) = fenics.TrialFunctions(W)
(tau, v, r_) = fenics.TestFunctions(W)
else:
W = fenics.FunctionSpace(mesh, DRT * CG)
# Define trial and test functions
(sigma, u) = fenics.TrialFunctions(W)
(tau, v) = fenics.TestFunctions(W)
f = s.source_function
g = s.g
# Define property field function
W_CG = fenics.FunctionSpace(mesh, "Lagrange", 1)
if s.ak is None:
ak = property_field.get_conductivity(W_CG, ak_values)
else:
ak = s.ak
# Define variational form
a = (fenics.dot(sigma, tau) + fenics.dot(ak * fenics.grad(u), tau) +
fenics.dot(sigma, fenics.grad(v))) * fenics.dx
L = -f * v * fenics.dx + g * v * fenics.ds
# L = 0
if s.integral_constraint:
# Lagrange multiplier? See above link.
a += r_ * u * fenics.dx + v * r * fenics.dx
# Define Dirichlet BC
bc = fenics.DirichletBC(W.sub(1), s.dirichlet_bc, s.gamma_dirichlet)
# Compute solution
w = fenics.Function(W)
fenics.solve(a == L, w, bc)
# fenics.solve(a == L, w)
if s.integral_constraint:
(sigma, u, r) = w.split()
else:
(sigma, u) = w.split()
x = u.compute_vertex_values(mesh)
x2 = sigma.compute_vertex_values(mesh)
p = x
pre = p.reshape((64, 64))
vx = x2[:4096].reshape((64, 64))
vy = x2[4096:].reshape((64, 64))
if obs_case == 1:
dd = np.zeros([8, 8])
pos = np.full((8 * 8, 2), 0)
col = [4, 12, 20, 28, 36, 44, 52, 60]
position = [4, 12, 20, 28, 36, 44, 52, 60]
for i in range(8):
for j in range(8):
row = position
pos[8 * i + j, :] = [col[i], row[j]]
dd[i, j] = pre[col[i], row[j]]
like = dd.reshape(8 * 8, )
return like, pre, vx, vy, ak_values, pos
b = fe.Expression(('0', '0'), degree=DEG) # forcing
nu = fe.Constant(2e-6)
rho = fe.Constant(1)
RE = 0.01
lmx = 1 # mixing length
dt = 0.1
# Re = 10 / 1e-4 = 1e5
V = fe.VectorElement("Lagrange", mesh.ufl_cell(), 2)
P = fe.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
NU = fe.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
if MODEL: M = fe.MixedElement([V, P, NU])
else: M = fe.MixedElement([V, P])
W = fe.FunctionSpace(mesh, M)
W0 = fe.Function(W)
We = fe.Function(W)
u0, p0 = fe.split(We)
#u0 = fe.Function((W0[0], W0[1]), 'Velocity000023.vtu')
#p0 = fe.Function(W0[2])
v, q = fe.TestFunctions(W)
#u, p = fe.split(W0)
u, p = (fe.as_vector((W0[0], W0[1])), W0[2])
#-------------------------------------------------------
# Defining essential/Dirichlet boundary conditions
# Step 1: Identify all boundary segments forming Gamma_d
#-------------------------------------------------------
# (-3., 2.5)
# |
# |
hessp=SR1(),
tol=1e-7,
constraints=volume_constraint,
bounds=((0, 1.0),) * A.dim(),
options={"verbose": 3, "gtol": 1e-7, "maxiter": 20},
)
q.assign(0.1)
res = minimize(
min_f,
res.x,
method="trust-constr",
jac=True,
hessp=SR1(),
tol=1e-7,
constraints=volume_constraint,
bounds=((0, 1.0),) * A.dim(),
options={"verbose": 3, "gtol": 1e-7, "maxiter": 100},
)
rho_opt_final = from_numpy(res.x, fenics.Function(A))
c = fenics.plot(rho_opt_final)
plt.colorbar(c)
plt.show()
rho_opt_file = fenics.XDMFFile(
fenics.MPI.comm_world, "output/control_solution_final.xdmf"
)
rho_opt_file.write(rho_opt_final)
def __init__(self, lagrangian, bcs_list, states, controls, adjoints,
config, riesz_scalar_products, control_constraints,
ksp_options, adjoint_ksp_options,
require_control_constraints):
"""Initializes the ControlFormHandler class.
Parameters
----------
lagrangian : cashocs._forms.Lagrangian
The lagrangian corresponding to the optimization problem.
bcs_list : list[list[dolfin.fem.dirichletbc.DirichletBC]]
The list of DirichletBCs for the state equation.
states : list[dolfin.function.function.Function]
The function that acts as the state variable.
controls : list[dolfin.function.function.Function]
The function that acts as the control variable.
adjoints : list[dolfin.function.function.Function]
The function that acts as the adjoint variable.
config : configparser.ConfigParser
The configparser object of the config file.
riesz_scalar_products : list[ufl.form.Form]
The UFL forms of the scalar products for the control variables.
control_constraints : list[list[dolfin.function.function.Function]]
The control constraints of the problem.
ksp_options : list[list[list[str]]]
The list of command line options for the KSP for the
state systems.
adjoint_ksp_options : list[list[list[str]]]
The list of command line options for the KSP for the
adjoint systems.
require_control_constraints : list[bool]
A list of boolean flags that indicates, whether the i-th control
has actual control constraints present.
"""
FormHandler.__init__(self, lagrangian, bcs_list, states, adjoints,
config, ksp_options, adjoint_ksp_options)
# Initialize the attributes from the arguments
self.controls = controls
self.riesz_scalar_products = riesz_scalar_products
self.control_constraints = control_constraints
self.require_control_constraints = require_control_constraints
self.control_dim = len(self.controls)
self.control_spaces = [x.function_space() for x in self.controls]
# Define the necessary functions
self.states_prime = [fenics.Function(V) for V in self.state_spaces]
self.adjoints_prime = [fenics.Function(V) for V in self.adjoint_spaces]
self.test_directions = [
fenics.Function(V) for V in self.control_spaces
]
self.trial_functions_control = [
fenics.TrialFunction(V) for V in self.control_spaces
]
self.test_functions_control = [
fenics.TestFunction(V) for V in self.control_spaces
]
self.temp = [fenics.Function(V) for V in self.control_spaces]
# Compute the necessary equations
self.__compute_gradient_equations()
if self.opt_algo == 'newton' or \
(self.opt_algo == 'pdas' and self.inner_pdas == 'newton'):
self.__compute_newton_forms()
# Initialize the scalar products
fenics_scalar_product_matrices = [
fenics.assemble(self.riesz_scalar_products[i], keep_diagonal=True)
for i in range(self.control_dim)
]
self.scalar_products_matrices = [
fenics.as_backend_type(fenics_scalar_product_matrices[i]).mat()
for i in range(self.control_dim)
]
copy_scalar_product_matrices = [
fenics_scalar_product_matrices[i].copy()
for i in range(self.control_dim)
]
[
copy_scalar_product_matrices[i].ident_zeros()
for i in range(self.control_dim)
]
self.riesz_projection_matrices = [
fenics.as_backend_type(copy_scalar_product_matrices[i]).mat()
for i in range(self.control_dim)
]
# Test for symmetry of the scalar products
for i in range(self.control_dim):
if not self.riesz_projection_matrices[i].isSymmetric():
if not self.riesz_projection_matrices[i].isSymmetric(1e-15):
if not (self.riesz_projection_matrices[i] -
self.riesz_projection_matrices[i].copy().transpose(
)).norm() / self.riesz_projection_matrices[i].norm(
) < 1e-15:
raise InputError(
'cashocs._forms.ControlFormHandler',
'riesz_scalar_products',
'Supplied scalar product form is not symmetric.')
def navierStokes(projectId, mesh, faceSets, boundarySets, config):
log("Navier Stokes Analysis has started")
# this is the default directory, when user request for download all files in this directory is being compressed and sent to the user
resultDir = "./Results/"
if len(config["steps"]) > 1:
return "more than 1 step is not supported yet"
# config is a dictionary containing all the user inputs for solver configurations
t_init = 0.0
t_final = float(config['steps'][0]["finalTime"])
t_num = int(config['steps'][0]["iterationNo"])
dt = ((t_final - t_init) / t_num)
t = t_init
#
# Viscosity coefficient.
#
nu = float(config['materials'][0]["viscosity"])
rho = float(config['materials'][0]["density"])
#
# Declare Finite Element Spaces
# do not use triangle directly
P2 = fn.VectorElement("P", mesh.ufl_cell(), 2)
P1 = fn.FiniteElement("P", mesh.ufl_cell(), 1)
TH = fn.MixedElement([P2, P1])
V = fn.VectorFunctionSpace(mesh, "P", 2)
Q = fn.FunctionSpace(mesh, "P", 1)
W = fn.FunctionSpace(mesh, TH)
#
# Declare Finite Element Functions
#
(u, p) = fn.TrialFunctions(W)
(v, q) = fn.TestFunctions(W)
w = fn.Function(W)
u0 = fn.Function(V)
p0 = fn.Function(Q)
#
# Macros needed for weak formulation.
#
def contract(u, v):
return fn.inner(fn.nabla_grad(u), fn.nabla_grad(v))
def b(u, v, w):
return 0.5 * (fn.inner(fn.dot(u, fn.nabla_grad(v)), w) -
fn.inner(fn.dot(u, fn.nabla_grad(w)), v))
# Define boundaries
bcs = []
for BC in config['BCs']:
if BC["boundaryType"] == "wall":
for edge in json.loads(BC["edges"]):
bcs.append(
fn.DirichletBC(W.sub(0),
fn.Constant((0.0, 0.0, 0.0)),
boundarySets,
int(edge),
method='topological'))
if BC["boundaryType"] == "inlet":
vel = json.loads(BC['value'])
for edge in json.loads(BC["edges"]):
bcs.append(
fn.DirichletBC(W.sub(0),
fn.Expression(
(str(vel[0]), str(vel[1]), str(vel[2])),
degree=2),
boundarySets,
int(edge),
method='topological'))
if BC["boundaryType"] == "outlet":
for edge in json.loads(BC["edges"]):
bcs.append(
fn.DirichletBC(W.sub(1),
fn.Constant(float(BC['value'])),
boundarySets,
int(edge),
method='topological'))
f = fn.Constant((0.0, 0.0, 0.0))
# weak form NSE
NSE = (1.0/dt)*fn.inner(u, v)*fn.dx + b(u0, u, v)*fn.dx + nu * \
contract(u, v)*fn.dx - fn.div(v)*p*fn.dx + q*fn.div(u)*fn.dx
LNSE = fn.inner(f, v) * fn.dx + (1. / dt) * fn.inner(u0, v) * fn.dx
velocity_file = fn.XDMFFile(resultDir + "/vel.xdmf")
pressure_file = fn.XDMFFile(resultDir + "/pressure.xdmf")
velocity_file.parameters["flush_output"] = True
velocity_file.parameters["functions_share_mesh"] = True
pressure_file.parameters["flush_output"] = True
pressure_file.parameters["functions_share_mesh"] = True
#
# code for projecting a boundary condition into a file for visualization
#
# for bc in bcs:
# bc.apply(w.vector())
# fn.File("para_plotting/bc.pvd") << w.sub(0)
for jj in range(0, t_num):
t = t + dt
# print('t = ' + str(t))
A, b = fn.assemble_system(NSE, LNSE, bcs)
fn.solve(A, w.vector(), b)
# fn.solve(NSE==LNSE,w,bcs)
fn.assign(u0, w.sub(0))
fn.assign(p0, w.sub(1))
# Save Solutions to Paraview File
if (jj % 20 == 0):
velocity_file.write(u0, t)
pressure_file.write(p0, t)
sendFile(projectId, resultDir + "vel.xdmf")
sendFile(projectId, resultDir + "vel.h5")
sendFile(projectId, resultDir + "pressure.xdmf")
sendFile(projectId, resultDir + "pressure.h5")
statusUpdate(projectId, "STARTED", {"progress": jj / t_num * 100})
def forward(lmin, pml, steps, Folder):
"""
Algorithm to solve the forward problem
*Input arguments
minl: Minimum lenght of element
pml: Size of the absorbing layer in km
steps: Number of timesteps
Folder: Folder name for results
*Output arguments
c: Field of propagation speed in domain
eta: Field of damping in domain
histrec: Historical data receivers
"""
histrec = ''
# Create and define function space
V = fn.FunctionSpace(mesh, "CG", 1)
w = fn.TrialFunction(V)
v = fn.TestFunction(V)
# Boundary conditions u0 = 0.0
bc = fn.DirichletBC(V, 0.0, fn.DomainBoundary())
# State variable and source space functions
w = fn.Function(V, name="Veloc")
w_ant1 = fn.Function(V)
w_ant2 = fn.Function(V)
q = fn.Function(V, name="Sourc")
# Velocity mapping
c_func = vel_c()
c = fn.interpolate(c_func, V)
velp_file = fn.File(Folder + "/Velp.pvd")
velp_file << c
# Damping mapping
damping_func = damping(c.vector().get_local().min())
eta = fn.interpolate(damping_func, V)
damp_file = fn.File(Folder + "/Damp.pvd")
damp_file << eta
# Current time we are solving for
t = 0.
# Number of timesteps solved
ntimestep = int(0)
# dt computation
dt = critical_dt(lmin, c.vector().get_local().max(), steps)
# Final time
T = steps*dt
print("Time step:", dt, "s")
# "Bilinear term"
def aterm(u, v, dt, c, eta):
termgrad = dt**2*fn.inner(c, c)*fn.inner(fn.grad(u), fn.grad(v))*fn.dx
termvare = (1.0 + dt*eta*fn.inner(c, c))*fn.inner(u, v)*fn.dx
termboun = (dt**2*c*c)*fn.inner(fn.grad(u), n)*v*fn.ds
return termgrad + termvare - termboun
# Source term
def lsourc(u_ant1, u_ant2, v, dt, c, q, eta):
termua1 = (-2. + dt*eta*fn.inner(c, c))*fn.inner(u_ant1, v)*fn.dx
termua2 = fn.inner(u_ant2, v)*fn.dx
termsou = (dt**2*fn.inner(c, c))*fn.inner(q, v)*fn.dx
return termua1 + termua2 - termsou
# Output file names
Exct = fn.File(Folder+"/Exct.pvd")
Wave = fn.File(Folder+"/Wave.pvd")
while True:
# Update the time it is solving for
print("Step: {0:1d} - Time: {1:1.4f}".format(ntimestep, t), "s")
# Ricker source for time t
ExcSource = RickerSource(t, tol=lmin/2)
q.assign(fn.interpolate(ExcSource, V))
# Time integration loop
if ntimestep < 1:
g = fn.interpolate(fn.Expression('0.0', degree=2), V)
w.assign(g) # assign initial guess for solver
w_ant1.assign(g) # assign initial guess for solver
w_ant2.assign(g) # assign initial guess for solver
else:
a = aterm(w, v, dt, c, eta)
L = lsourc(w_ant1, w_ant2, v, dt, c, q, eta)
#solve equation
fn.solve(a + L == 0, w, bcs=bc, solver_parameters={"newton_solver":
{"maximum_iterations": forward_max_it, "absolute_tolerance": forward_tol,
"relative_tolerance": relativ_tol}})
# Cycling the variables
w_ant2.assign(w_ant1)
w_ant1.assign(w)
#Output files
# histrec += Receiver(t, posrec, pml, w)
Exct << (q, t)
Wave << (w, t)
t += dt
ntimestep += 1
if t >= T:
break
return c, eta, histrec
bcU = fn.DirichletBC(Hh.sub(0), u_g, bdry, 31)
bcP = fn.DirichletBC(Hh.sub(2), p_g, bdry, 32)
bcs = [bcU, bcP]
# ******** Weak forms ********** #
PLeft = 2*mu*fn.inner(strain(u),strain(v)) * fn.dx \
- fn.div(v) * phi * fn.dx \
+ (c0/alpha + 1.0/lmbda)* p * q * fn.dx \
+ kappa/(alpha*nu) * fn.dot(fn.grad(p),fn.grad(q)) * fn.dx \
- 1.0/lmbda * phi * q * fn.dx \
- fn.div(u) * psi * fn.dx \
+ 1.0/lmbda * psi * p * fn.dx \
- 1.0/lmbda * phi * psi * fn.dx
PRight = fn.dot(f, v) * fn.dx + fn.dot(h_g, v) * ds(32)
Sol = fn.Function(Hh)
# solve
fn.solve(PLeft == PRight, Sol, bcs)
u, phi, p = Sol.split()
u.rename("u", "u")
fileU << u
p.rename("p", "p")
fileP << p
phi.rename("phi", "phi")
filePHI << phi
def __init__(self, state_forms, bcs_list, cost_functional_form, states, controls, adjoints, config=None,
riesz_scalar_products=None, control_constraints=None, initial_guess=None, ksp_options=None,
adjoint_ksp_options=None, desired_weights=None):
r"""This is used to generate all classes and functionalities. First ensures
consistent input, afterwards, the solution algorithm is initialized.
Parameters
----------
state_forms : ufl.form.Form or list[ufl.form.Form]
The weak form of the state equation (user implemented). Can be either
a single UFL form, or a (ordered) list of UFL forms.
bcs_list : list[dolfin.fem.dirichletbc.DirichletBC] or list[list[dolfin.fem.dirichletbc.DirichletBC]] or dolfin.fem.dirichletbc.DirichletBC or None
The list of DirichletBC objects describing Dirichlet (essential) boundary conditions.
If this is ``None``, then no Dirichlet boundary conditions are imposed.
cost_functional_form : ufl.form.Form or list[ufl.form.Form]
UFL form of the cost functional.
states : dolfin.function.function.Function or list[dolfin.function.function.Function]
The state variable(s), can either be a :py:class:`fenics.Function`, or a list of these.
controls : dolfin.function.function.Function or list[dolfin.function.function.Function]
The control variable(s), can either be a :py:class:`fenics.Function`, or a list of these.
adjoints : dolfin.function.function.Function or list[dolfin.function.function.Function]
The adjoint variable(s), can either be a :py:class:`fenics.Function`, or a (ordered) list of these.
config : configparser.ConfigParser or None
The config file for the problem, generated via :py:func:`cashocs.create_config`.
Alternatively, this can also be ``None``, in which case the default configurations
are used, except for the optimization algorithm. This has then to be specified
in the :py:meth:`solve <cashocs.OptimalControlProblem.solve>` method. The
default is ``None``.
riesz_scalar_products : None or ufl.form.Form or list[ufl.form.Form], optional
The scalar products of the control space. Can either be None, a single UFL form, or a
(ordered) list of UFL forms. If ``None``, the :math:`L^2(\Omega)` product is used.
(default is ``None``).
control_constraints : None or list[dolfin.function.function.Function] or list[float] or list[list[dolfin.function.function.Function]] or list[list[float]], optional
Box constraints posed on the control, ``None`` means that there are none (default is ``None``).
The (inner) lists should contain two elements of the form ``[u_a, u_b]``, where ``u_a`` is the lower,
and ``u_b`` the upper bound.
initial_guess : list[dolfin.function.function.Function], optional
List of functions that act as initial guess for the state variables, should be valid input for :py:func:`fenics.assign`.
Defaults to ``None``, which means a zero initial guess.
ksp_options : list[list[str]] or list[list[list[str]]] or None, optional
A list of strings corresponding to command line options for PETSc,
used to solve the state systems. If this is ``None``, then the direct solver
mumps is used (default is ``None``).
adjoint_ksp_options : list[list[str]] or list[list[list[str]]] or None
A list of strings corresponding to command line options for PETSc,
used to solve the adjoint systems. If this is ``None``, then the same options
as for the state systems are used (default is ``None``).
Examples
--------
Examples how to use this class can be found in the :ref:`tutorial <tutorial_index>`.
"""
OptimizationProblem.__init__(self, state_forms, bcs_list, cost_functional_form, states, adjoints, config, initial_guess, ksp_options, adjoint_ksp_options, desired_weights)
### Overloading, such that we do not have to use lists for a single state and a single control
### controls
try:
if type(controls) == list and len(controls) > 0:
for i in range(len(controls)):
if controls[i].__module__ == 'dolfin.function.function' and type(controls[i]).__name__ == 'Function':
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'controls', 'controls have to be fenics Functions.')
self.controls = controls
elif controls.__module__ == 'dolfin.function.function' and type(controls).__name__ == 'Function':
self.controls = [controls]
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'controls', 'Type of controls is wrong.')
except:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'controls', 'Type of controls is wrong.')
self.control_dim = len(self.controls)
### riesz_scalar_products
if riesz_scalar_products is None:
dx = fenics.Measure('dx', self.controls[0].function_space().mesh())
self.riesz_scalar_products = [fenics.inner(fenics.TrialFunction(self.controls[i].function_space()), fenics.TestFunction(self.controls[i].function_space())) * dx
for i in range(len(self.controls))]
else:
try:
if type(riesz_scalar_products)==list and len(riesz_scalar_products) > 0:
for i in range(len(riesz_scalar_products)):
if riesz_scalar_products[i].__module__== 'ufl.form' and type(riesz_scalar_products[i]).__name__== 'Form':
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'riesz_scalar_products', 'riesz_scalar_products have to be ufl forms')
self.riesz_scalar_products = riesz_scalar_products
elif riesz_scalar_products.__module__== 'ufl.form' and type(riesz_scalar_products).__name__== 'Form':
self.riesz_scalar_products = [riesz_scalar_products]
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'riesz_scalar_products', 'riesz_scalar_products have to be ufl forms')
except:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'riesz_scalar_products', 'riesz_scalar_products have to be ufl forms')
### control_constraints
if control_constraints is None:
self.control_constraints = []
for control in self.controls:
u_a = fenics.Function(control.function_space())
u_a.vector()[:] = float('-inf')
u_b = fenics.Function(control.function_space())
u_b.vector()[:] = float('inf')
self.control_constraints.append([u_a, u_b])
else:
try:
if type(control_constraints) == list and len(control_constraints) > 0:
if type(control_constraints[0]) == list:
for i in range(len(control_constraints)):
if type(control_constraints[i]) == list and len(control_constraints[i]) == 2:
for j in range(2):
if type(control_constraints[i][j]) in [float, int]:
pass
elif control_constraints[i][j].__module__ == 'dolfin.function.function' and type(control_constraints[i][j]).__name__ == 'Function':
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints',
'control_constraints has to be a list containing upper and lower bounds')
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints',
'control_constraints has to be a list containing upper and lower bounds')
self.control_constraints = control_constraints
elif (type(control_constraints[0]) in [float, int] or (control_constraints[0].__module__ == 'dolfin.function.function' and type(control_constraints[0]).__name__=='Function')) \
and (type(control_constraints[1]) in [float, int] or (control_constraints[1].__module__ == 'dolfin.function.function' and type(control_constraints[1]).__name__=='Function')):
self.control_constraints = [control_constraints]
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints',
'control_constraints has to be a list containing upper and lower bounds')
except:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints',
'control_constraints has to be a list containing upper and lower bounds')
# recast floats into functions for compatibility
temp_constraints = self.control_constraints[:]
self.control_constraints = []
for idx, pair in enumerate(temp_constraints):
if type(pair[0]) in [float, int]:
lower_bound = fenics.Function(self.controls[idx].function_space())
lower_bound.vector()[:] = pair[0]
elif pair[0].__module__ == 'dolfin.function.function' and type(pair[0]).__name__ == 'Function':
lower_bound = pair[0]
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints', 'Wrong type for the control constraints')
if type(pair[1]) in [float, int]:
upper_bound = fenics.Function(self.controls[idx].function_space())
upper_bound.vector()[:] = pair[1]
elif pair[1].__module__ == 'dolfin.function.function' and type(pair[1]).__name__ == 'Function':
upper_bound = pair[1]
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints', 'Wrong type for the control constraints')
self.control_constraints.append([lower_bound, upper_bound])
### Check whether the control constraints are feasible, and whether they are actually present
self.require_control_constraints = [False for i in range(self.control_dim)]
for idx, pair in enumerate(self.control_constraints):
if not np.alltrue(pair[0].vector()[:] < pair[1].vector()[:]):
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints',
'The lower bound must always be smaller than the upper bound for the control_constraints.')
if np.max(pair[0].vector()[:]) == float('-inf') and np.min(pair[1].vector()[:]) == float('inf'):
# no control constraint for this component
pass
else:
self.require_control_constraints[idx] = True
control_element = self.controls[idx].ufl_element()
if control_element.family() == 'Mixed':
for j in range(control_element.value_size()):
sub_elem = control_element.extract_component(j)[1]
if sub_elem.family() == 'Real' or (sub_elem.family() == 'Lagrange' and sub_elem.degree() == 1) \
or (sub_elem.family() == 'Discontinuous Lagrange' and sub_elem.degree() == 0):
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'controls',
'Control constraints are only implemented for linear Lagrange, constant Discontinuous Lagrange, and Real elements.')
else:
if control_element.family() == 'Real' or (control_element.family() == 'Lagrange' and control_element.degree() == 1) \
or (control_element.family() == 'Discontinuous Lagrange' and control_element.degree() == 0):
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'controls',
'Control constraints are only implemented for linear Lagrange, constant Discontinuous Lagrange, and Real elements.')
if not len(self.riesz_scalar_products) == self.control_dim:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'riesz_scalar_products', 'Length of controls does not match')
if not len(self.control_constraints) == self.control_dim:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints', 'Length of controls does not match')
### end overloading
self.form_handler = ControlFormHandler(self.lagrangian, self.bcs_list, self.states, self.controls, self.adjoints, self.config,
self.riesz_scalar_products, self.control_constraints, self.ksp_options, self.adjoint_ksp_options,
self.require_control_constraints)
self.state_spaces = self.form_handler.state_spaces
self.control_spaces = self.form_handler.control_spaces
self.adjoint_spaces = self.form_handler.adjoint_spaces
self.projected_difference = [fenics.Function(V) for V in self.control_spaces]
self.state_problem = StateProblem(self.form_handler, self.initial_guess)
self.adjoint_problem = AdjointProblem(self.form_handler, self.state_problem)
self.gradient_problem = GradientProblem(self.form_handler, self.state_problem, self.adjoint_problem)
self.algorithm = _optimization_algorithm_configuration(self.config)
self.reduced_cost_functional = ReducedCostFunctional(self.form_handler, self.state_problem)
self.gradients = self.gradient_problem.gradients
self.objective_value = 1.0
def __compute_shape_gradient_forms(self):
"""Calculates the necessary left-hand-sides for the shape gradient problem.
Returns
-------
None
"""
self.shape_bdry_def = json.loads(
self.config.get('ShapeGradient', 'shape_bdry_def', fallback='[]'))
if not type(self.shape_bdry_def) == list:
raise ConfigError('ShapeGradient', 'shape_bdry_def',
'The input has to be a list.')
# if not len(self.shape_bdry_def) > 0:
# raise ConfigError('ShapeGradient', 'shape_bdry_def','The input must not be empty.')
self.shape_bdry_fix = json.loads(
self.config.get('ShapeGradient', 'shape_bdry_fix', fallback='[]'))
if not type(self.shape_bdry_def) == list:
raise ConfigError('ShapeGradient', 'shape_bdry_fix',
'The input has to be a list.')
self.shape_bdry_fix_x = json.loads(
self.config.get('ShapeGradient', 'shape_bdry_fix_x',
fallback='[]'))
if not type(self.shape_bdry_fix_x) == list:
raise ConfigError('ShapeGradient', 'shape_bdry_fix_x',
'The input has to be a list.')
self.shape_bdry_fix_y = json.loads(
self.config.get('ShapeGradient', 'shape_bdry_fix_y',
fallback='[]'))
if not type(self.shape_bdry_fix_y) == list:
raise ConfigError('ShapeGradient', 'shape_bdry_fix_y',
'The input has to be a list.')
self.shape_bdry_fix_z = json.loads(
self.config.get('ShapeGradient', 'shape_bdry_fix_z',
fallback='[]'))
if not type(self.shape_bdry_fix_z) == list:
raise ConfigError('ShapeGradient', 'shape_bdry_fix_z',
'The input has to be a list.')
self.bcs_shape = create_bcs_list(
self.deformation_space,
fenics.Constant([0] *
self.deformation_space.ufl_element().value_size()),
self.boundaries, self.shape_bdry_fix)
self.bcs_shape += create_bcs_list(self.deformation_space.sub(0),
fenics.Constant(0.0),
self.boundaries,
self.shape_bdry_fix_x)
self.bcs_shape += create_bcs_list(self.deformation_space.sub(1),
fenics.Constant(0.0),
self.boundaries,
self.shape_bdry_fix_y)
if self.deformation_space.num_sub_spaces() == 3:
self.bcs_shape += create_bcs_list(self.deformation_space.sub(2),
fenics.Constant(0.0),
self.boundaries,
self.shape_bdry_fix_z)
self.CG1 = fenics.FunctionSpace(self.mesh, 'CG', 1)
self.DG0 = fenics.FunctionSpace(self.mesh, 'DG', 0)
if self.shape_scalar_product is None:
# Use the default linear elasticity approach
self.mu_lame = fenics.Function(self.CG1)
self.lambda_lame = self.config.getfloat('ShapeGradient',
'lambda_lame',
fallback=0.0)
self.damping_factor = self.config.getfloat('ShapeGradient',
'damping_factor',
fallback=0.0)
if self.config.getboolean('ShapeGradient',
'inhomogeneous',
fallback=False):
self.volumes = fenics.project(fenics.CellVolume(self.mesh),
self.DG0)
vol_max = np.max(np.abs(self.volumes.vector()[:]))
self.volumes.vector()[:] /= vol_max
else:
self.volumes = fenics.Constant(1.0)
def eps(u):
return fenics.Constant(0.5) * (fenics.grad(u) +
fenics.grad(u).T)
trial = fenics.TrialFunction(self.deformation_space)
test = fenics.TestFunction(self.deformation_space)
self.riesz_scalar_product = fenics.Constant(2)*self.mu_lame/self.volumes*fenics.inner(eps(trial), eps(test))*self.dx \
+ fenics.Constant(self.lambda_lame)/self.volumes*fenics.div(trial)*fenics.div(test)*self.dx \
+ fenics.Constant(self.damping_factor)/self.volumes*fenics.inner(trial, test)*self.dx
else:
# Use the scalar product supplied by the user
self.riesz_scalar_product = self.shape_scalar_product
def __init__(self, optimization_problem):
"""Parent class for the optimization methods implemented in cashocs.optimization.methods
Parameters
----------
optimization_problem : cashocs.ShapeOptimizationProblem
the optimization problem
"""
self.line_search_broken = False
self.has_curvature_info = False
self.optimization_problem = optimization_problem
self.form_handler = self.optimization_problem.form_handler
self.state_problem = self.optimization_problem.state_problem
self.config = self.state_problem.config
self.adjoint_problem = self.optimization_problem.adjoint_problem
self.shape_gradient_problem = self.optimization_problem.shape_gradient_problem
self.gradient = self.shape_gradient_problem.gradient
self.cost_functional = self.optimization_problem.reduced_cost_functional
self.search_direction = fenics.Function(
self.form_handler.deformation_space)
if self.config.getboolean('Mesh', 'remesh', fallback=False):
self.iteration = self.optimization_problem.temp_dict[
'OptimizationRoutine'].get('iteration_counter', 0)
else:
self.iteration = 0
self.objective_value = 1.0
self.gradient_norm_initial = 1.0
self.relative_norm = 1.0
self.stepsize = 1.0
self.converged = False
self.converged_reason = 0
self.output_dict = dict()
try:
self.output_dict[
'cost_function_value'] = self.optimization_problem.temp_dict[
'output_dict']['cost_function_value']
self.output_dict[
'gradient_norm'] = self.optimization_problem.temp_dict[
'output_dict']['gradient_norm']
self.output_dict['stepsize'] = self.optimization_problem.temp_dict[
'output_dict']['stepsize']
self.output_dict[
'MeshQuality'] = self.optimization_problem.temp_dict[
'output_dict']['MeshQuality']
except (TypeError, KeyError):
self.output_dict['cost_function_value'] = []
self.output_dict['gradient_norm'] = []
self.output_dict['stepsize'] = []
self.output_dict['MeshQuality'] = []
self.verbose = self.config.getboolean('Output',
'verbose',
fallback=True)
self.save_results = self.config.getboolean('Output',
'save_results',
fallback=True)
self.rtol = self.config.getfloat('OptimizationRoutine',
'rtol',
fallback=1e-3)
self.atol = self.config.getfloat('OptimizationRoutine',
'atol',
fallback=0.0)
self.maximum_iterations = self.config.getint('OptimizationRoutine',
'maximum_iterations',
fallback=100)
self.soft_exit = self.config.getboolean('OptimizationRoutine',
'soft_exit',
fallback=False)
self.save_pvd = self.config.getboolean('Output',
'save_pvd',
fallback=False)
self.result_dir = self.config.get('Output',
'result_dir',
fallback='./')
if self.save_pvd:
self.state_pvd_list = []
for i in range(self.form_handler.state_dim):
if self.form_handler.state_spaces[i].num_sub_spaces() > 0:
self.state_pvd_list.append([])
for j in range(self.form_handler.state_spaces[i].
num_sub_spaces()):
if self.optimization_problem.mesh_handler.do_remesh:
self.state_pvd_list[i].append(
fenics.
File(self.result_dir + '/pvd/remesh_' + format(
self.optimization_problem.temp_dict.get(
'remesh_counter', 0), 'd') +
'_state_' + str(i) + '_' + str(j) +
'.pvd'))
else:
self.state_pvd_list[i].append(
fenics.File(self.result_dir + '/pvd/state_' +
str(i) + '_' + str(j) + '.pvd'))
else:
if self.optimization_problem.mesh_handler.do_remesh:
self.state_pvd_list.append(
fenics.File(self.result_dir + '/pvd/remesh_' +
format(
self.optimization_problem.temp_dict
.get('remesh_counter', 0), 'd') +
'_state_' + str(i) + '.pvd'))
else:
self.state_pvd_list.append(
fenics.File(self.result_dir + '/pvd/state_' +
str(i) + '.pvd'))
mu = 1 # kinematic viscosity rho = 1 # density # Create mesh and define function spaces SEPARATELY for pressure and velocity spaces. mesh = fs.UnitSquareMesh(16, 16) V = fs.VectorFunctionSpace(mesh, 'P', 2) # velocity space, a vector space Q = fs.FunctionSpace(mesh, 'P', 1) # pressure space, a scalar space # Define trial and test functions u = fs.TrialFunction(V) # in velocity space v = fs.TestFunction(V) p = fs.TrialFunction(Q) # in pressure space q = fs.TestFunction(Q) # Define functions for solutions at previous and current time steps u_n = fs.Function(V) # u^(n) u_ = fs.Function(V) # u^(n+1) p_n = fs.Function(Q) # p^(n) p_ = fs.Function(Q) # p^(n+1) # Define boundaries inflow = 'near(x[0], 0)' outflow = 'near(x[0], 1)' walls = 'near(x[1], 0) || near (x[1], 1)' # Define boundary bonditions bcu_noslip = fs.DirichletBC(V, fs.Constant((0,0)), walls) bcp_inflow = fs.DirichletBC(Q, fs.Constant(8), inflow) bcp_outflow = fs.DirichletBC(Q, fs.Constant(0), outflow) bcu = [bcu_noslip] bcp = [bcp_inflow, bcp_outflow]
def demo16(self, permeability, obs_case=1):
"""This demo program solves the mixed formulation of Poisson's
equation:
sigma + grad(u) = 0 in Omega
div(sigma) = f in Omega
du/dn = g on Gamma_N
u = u_D on Gamma_D
The corresponding weak (variational problem)
<sigma, tau> + <grad(u), tau> = 0
for all tau
- <sigma, grad(v)> = <f, v> + <g, v>
for all v
is solved using DRT (Discontinuous Raviart-Thomas) elements
of degree k for (sigma, tau) and CG (Lagrange) elements
of degree k + 1 for (u, v) for k >= 1.
"""
mesh = UnitSquareMesh(15, 15)
ak_values = permeability
flux_order = 3
s = scenarios.darcy_problem_1()
DRT = fenics.FiniteElement("DRT", mesh.ufl_cell(), flux_order)
# Lagrange
CG = fenics.FiniteElement("CG", mesh.ufl_cell(), flux_order + 1)
if s.integral_constraint:
# From https://fenicsproject.org/qa/14184/how-to-solve-linear-system-with-constaint
R = fenics.FiniteElement("R", mesh.ufl_cell(), 0)
W = fenics.FunctionSpace(mesh, fenics.MixedElement([DRT, CG, R]))
# Define trial and test functions
(sigma, u, r) = fenics.TrialFunctions(W)
(tau, v, r_) = fenics.TestFunctions(W)
else:
W = fenics.FunctionSpace(mesh, DRT * CG)
# Define trial and test functions
(sigma, u) = fenics.TrialFunctions(W)
(tau, v) = fenics.TestFunctions(W)
f = s.source_function
g = s.g
# Define property field function
W_CG = fenics.FunctionSpace(mesh, "Lagrange", 1)
if s.ak is None:
ak = property_field.get_conductivity(W_CG, ak_values)
else:
ak = s.ak
# Define variational form
a = (fenics.dot(sigma, tau) + fenics.dot(ak * fenics.grad(u), tau) +
fenics.dot(sigma, fenics.grad(v))) * fenics.dx
L = -f * v * fenics.dx + g * v * fenics.ds
# L = 0
if s.integral_constraint:
# Lagrange multiplier? See above link.
a += r_ * u * fenics.dx + v * r * fenics.dx
# Define Dirichlet BC
bc = fenics.DirichletBC(W.sub(1), s.dirichlet_bc, s.gamma_dirichlet)
# Compute solution
w = fenics.Function(W)
fenics.solve(a == L, w, bc)
# fenics.solve(a == L, w)
if s.integral_constraint:
(sigma, u, r) = w.split()
else:
(sigma, u) = w.split()
x = u.compute_vertex_values(mesh)
x2 = sigma.compute_vertex_values(mesh)
p = x
pre = p.reshape((16, 16))
vx = x2[:256].reshape((16, 16))
vy = x2[256:].reshape((16, 16))
if obs_case == 1:
dd = np.zeros([8, 8])
pos = np.full((8 * 8, 2), 0)
col = [1, 3, 5, 7, 9, 11, 13, 15]
position = [1, 3, 5, 7, 9, 11, 13, 15]
for i in range(8):
for j in range(8):
row = position
pos[8 * i + j, :] = [col[i], row[j]]
dd[i, j] = pre[col[i], row[j]]
like = dd.reshape(8 * 8, )
return like, pre, vx, vy, ak_values, pos
def solve_cell_problems(self, method='regularized', plot=False):
"""
Solve the cell problems for the given PDE by changing the geometry to exclude the zone in the middle
:return:
"""
class PeriodicBoundary(fe.SubDomain):
# Left boundary is "target domain" G
def inside(self, x, on_boundary):
par = MembraneSimulator.w / MembraneSimulator.length
tol = 1E-4
return on_boundary and x[1] >= -tol and x[1] <= tol
# Map right boundary (H) to left boundary (G)
def map(self, x, y):
y[1] = x[1] + 2 * self.eta / MembraneSimulator.length
y[0] = x[0]
mesh_size = 50
# Domain
if method == 'circle':
self.obstacle_radius = self.eta / MembraneSimulator.length / 2
box_begin_point = fe.Point(-self.w / MembraneSimulator.length, 0)
box_end_point = fe.Point(self.w / MembraneSimulator.length,
2 * self.eta / MembraneSimulator.length)
box = mshr.Rectangle(box_begin_point, box_end_point)
cell = box - mshr.Circle(
fe.Point(0, self.eta / MembraneSimulator.length),
self.obstacle_radius, mesh_size)
self.cell_mesh = mshr.generate_mesh(cell, mesh_size)
diff_coef = fe.Constant(
((0, 0), (0, self.D[1, 1]))) # limit for regularisation below.
# elif method == 'diff_coef':
# print("Haha this is going to crash. Also it is horrible in accuracy")
# self.cell_mesh = fe.RectangleMesh(fe.Point(-self.w / MembraneSimulator.length, 0),
# fe.Point(self.w / MembraneSimulator.length,
# 2 * self.eta / MembraneSimulator.length), mesh_size, mesh_size)
# diff_coef = fe.Expression(
# (('0', '0'), ('0', 'val * ((x[0] - c_x)*(x[0] - c_x) + (x[1] - c_y)*(x[1] - c_y) > r*r)')),
# val=(4 * self.ref_T * MembraneSimulator.d2 / MembraneSimulator.length ** 2), c_x=0,
# c_y=(self.eta / MembraneSimulator.length),
# r=self.obstacle_radius, degree=2, domain=self.cell_mesh)
elif method == 'regularized':
box_begin_point = fe.Point(-self.w / MembraneSimulator.length, 0)
box_end_point = fe.Point(self.w / MembraneSimulator.length,
2 * self.eta / MembraneSimulator.length)
box = mshr.Rectangle(box_begin_point, box_end_point)
obstacle_begin_point = fe.Point(
-self.w / MembraneSimulator.length * self.obs_length,
self.eta / MembraneSimulator.length * (1 - self.obs_height))
obstacle_end_point = fe.Point(
self.w / MembraneSimulator.length * self.obs_length,
self.eta / MembraneSimulator.length * (1 + self.obs_height))
obstacle = mshr.Rectangle(obstacle_begin_point, obstacle_end_point)
cell = box - obstacle
self.cell_mesh = mshr.generate_mesh(cell, mesh_size)
diff_coef = fe.Constant(((self.obs_ratio * self.D[0, 0], 0),
(0, self.D[1, 1]))) # defect matrix.
else:
raise ValueError("%s not a valid method to solve cell problem" %
method)
self.cell_fs = fe.FunctionSpace(self.cell_mesh, 'P', 2)
self.cell_solutions = [
fe.Function(self.cell_fs),
fe.Function(self.cell_fs)
]
w = fe.TrialFunction(self.cell_fs)
phi = fe.TestFunction(self.cell_fs)
scaled_unit_vectors = [
fe.Constant((1. / np.sqrt(self.obs_ratio), 0.)),
fe.Constant((0., 1.))
]
for i in range(2):
weak_form = fe.dot(
diff_coef *
(fe.grad(w) + scaled_unit_vectors[i]), fe.grad(phi)) * fe.dx
print("Solving cell problem")
bc = fe.DirichletBC(self.cell_fs, fe.Constant(0),
MembraneSimulator.cell_boundary)
if i == 0:
# Periodicity is applied automatically
bc = None
fe.solve(
fe.lhs(weak_form) == fe.rhs(weak_form), self.cell_solutions[i],
bc)
if plot:
plt.rc('text', usetex=True)
f = fe.plot(self.cell_solutions[i])
plt.colorbar(f)
plt.title(r'Solution to cell problem $w_%d$' % (i + 1))
plt.xlabel(r'$Y_1$')
plt.ylabel(r'$Y_2$')
print("Cell solution")
print(np.min(self.cell_solutions[i].vector().get_local()),
np.max(self.cell_solutions[i].vector().get_local()))
plt.show()
def _load_fenics_function(path):
V = fn.FunctionSpace(mesh, element_type, element_degree)
return fn.Function(V, path)
MODEL = False # flag to use SA model
b = fe.Expression(('0', '0'), degree=DEG) # forcing
nu = fe.Constant(1e-2)
rho = fe.Constant(1)
RE = 50
lmx = 1 # mixing length :)
# Re = 10 / 1e-4 = 1e5
V = fe.VectorElement("Lagrange", mesh.ufl_cell(), 2)
P = fe.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
NU = fe.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
if MODEL: M = fe.MixedElement([V, P, NU])
else: M = fe.MixedElement([V, P])
W = fe.FunctionSpace(mesh, M)
W0 = fe.Function(W)
if MODEL:
(v, q, nu_test) = fe.TestFunctions(W)
(u, p, nu_trial) = fe.split(W0)
else:
(
v,
q,
) = fe.TestFunctions(W)
(
u,
p,
) = fe.split(W0)
nu_trial = fe.Constant(5) # artificial viscosity!!!
fv1 = fe.Constant(1)
Mixed = fn.MixedElement(Elem)
V = fn.FunctionSpace(mesh, Mixed)
# %%
# define potentials and concentrations
u_GND = fn.Expression('0', degree=2) #Ground
u_DD = fn.Expression('0.5', degree=2) #pontential
c_avg = fn.Expression('0.0001', degree=2) #average concentration
# set boundary conditions
bcs = []
bcs += [fn.DirichletBC(V.sub(0), u_DD, mf, 3)]
bcs += [fn.DirichletBC(V.sub(0), u_GND, mf, 1)]
# define problem
UC = fn.Function(V)
uc = fn.split(UC) # trial function potential concentration lagrange multi
u, c, lam = uc[0], uc[1:M + 1], uc[M + 1:]
VW = fn.TestFunctions(
V) # test function potential concentration lagrange multi
v, w, mu = VW[0], VW[1:M + 1], VW[M + 1:]
#lets try rot
r = fn.Expression('x[0]', degree=0)
# changing concentrations charges
Rho = 0
for i in range(M):
if i % 2:
Rho += -c[i]
# Volumetric Load
q = -10.0 / t
b = fn.Constant((0.0, q))
def Left_boundary(x, on_boundary):
return on_boundary and abs(x[0]) < fn.DOLFIN_EPS
u_L = fn.Constant((0.0, 0.0))
bcs = [fn.DirichletBC(V, u_L, Left_boundary)]
# Forward problem solution
@build_jax_solve_eval((fn.Function(C), ))
def forward(x):
u = fn.TrialFunction(V)
w = fn.TestFunction(V)
sigma = lmbda * tr(sym(grad(u))) * Identity(2) + 2 * G * sym(
grad(u)) # Stress
R = simp(x) * inner(sigma, grad(w)) * dx - dot(b, w) * dx
a, L = ufl.lhs(R), ufl.rhs(R)
u = fn.Function(V)
F = L - ufl.action(a, u)
fn.solve(a == L, u, bcs)
return u, F, bcs
@build_jax_assemble_eval((fn.Function(V), fn.Function(C)))
def eval_cost(u, x):
boundaries = fe.MeshFunction("size_t", mesh, "mesh/beam_facet_region.xml")
# Redefine the integration measures
dxp = fe.Measure('dx', domain=mesh, subdomain_data=subdomains)
dsp = fe.Measure('ds', domain=mesh, subdomain_data=boundaries)
##################### FINITE ELEMENT SPACES ##################################
# Finite element spaces
W = fe.FunctionSpace(mesh, 'P', 1)
V = fe.VectorFunctionSpace(mesh, 'P', 1)
Z = fe.TensorFunctionSpace(mesh, 'P', 1)
# Finite element functions
du = fe.TrialFunction(V)
v = fe.TestFunction(V)
u = fe.Function(V)
######################### PROBLEM PARAMS ######################################
# Material properties
materials = {
1: [210e9, 0.33],
}
# Load multipliers
p = 500000.
q = p * 100.
######################## BOUNDARY CONDITIONS #################################
# Define the boundary conditions
boundary_conditions = {
2: fe.Constant((0., 0., 0.)),
return on_boundary and fen.near(x[1], edge, tol)
def boundary_B(x, on_boundary):
return on_boundary and fen.near(x[1], -edge, tol)
bc_L = fen.DirichletBC(V, phi_L, boundary_L)
bc_R = fen.DirichletBC(V, phi_R, boundary_R)
bc_H = fen.DirichletBC(V, phi_H, boundary_H)
bc_B = fen.DirichletBC(V, phi_B, boundary_B)
bc = [bc_L, bc_R, bc_H, bc_B]
# -------------Define variational problem------------
phi = fen.Function(V)
v = fen.TestFunction(V)
pi = np.pi
rho_sun = fen.Expression(
'pow(pow(x[0],2)+pow(x[1],2),0.5)<=rad_sun ? 3*m_sun/(4*pi*pow(rad_sun,3)): 0',
degree=2,
rad_sun=rad_sun,
m_sun=m_sun,
pi=pi)
J = fen.Expression('pow(x[0]*x[0] + x[1]*x[1],0.5)', degree=2)
Func = (-J * phi.dx(0) * v.dx(0) - J * phi.dx(1) * v.dx(1) -
J * 4 * np.pi * rho_sun * v) * fen.dx
# Define Boundaries
left = fe.CompiledSubDomain("near(x[0], side) && on_boundary", side=0.0)
right = fe.CompiledSubDomain("near(x[0], side) && on_boundary", side=1.0)
# Define Dirichlet boundary (x = 0 or x = 1)
u_left = fe.Expression(("0.0", "0.0", "0.0"), element=element_3)
u_right = fe.Expression(("0.0", "0.0", "0.0"), element=element_3)
p_left = fe.Constant(0.)
# Define acting force
b = fe.Constant((0.0, 0.0, 0.0)) # Body force per unit volume
t_bar = fe.Constant((0.0, 0.0, 0.0)) # Traction force on the boundary
# Define test and trial functions
w = fe.Function(V) # most recently computed solution
(u, p) = fe.split(w)
(v, q) = fe.TestFunctions(V)
dw = fe.TrialFunction(V)
# Kinematics
d = u.geometric_dimension()
I = fe.Identity(d) # Identity tensor
F = fe.variable(I + grad(u)) # Deformation gradient
C = fe.variable(F.T * F) # Right Cauchy-Green tensor
J = fe.det(C)
DE = lambda v: 0.5 * (F.T * grad(v) + grad(v).T * F)
a_0 = fe.as_vector([[1.0], [0.], [0.]])
def __init__(self, form_handler, gradient_problem):
"""Initializes self.
Parameters
----------
form_handler : cashocs._forms.ControlFormHandler
The FormHandler object for the optimization problem.
gradient_problem : cashocs._pde_problems.GradientProblem
The GradientProblem object (this is needed for the computation
of the Hessian).
"""
self.form_handler = form_handler
self.gradient_problem = gradient_problem
self.config = self.form_handler.config
self.gradients = self.gradient_problem.gradients
self.inner_newton = self.config.get('AlgoTNM',
'inner_newton',
fallback='cr')
self.max_it_inner_newton = self.config.getint('AlgoTNM',
'max_it_inner_newton',
fallback=50)
# TODO: Add absolute tolerance, too
self.inner_newton_tolerance = self.config.getfloat(
'AlgoTNM', 'inner_newton_tolerance', fallback=1e-15)
self.test_directions = self.form_handler.test_directions
self.residual = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.delta_control = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.state_dim = self.form_handler.state_dim
self.control_dim = self.form_handler.control_dim
self.temp1 = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.temp2 = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.p = [fenics.Function(V) for V in self.form_handler.control_spaces]
self.p_prev = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.p_pprev = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.s = [fenics.Function(V) for V in self.form_handler.control_spaces]
self.s_prev = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.s_pprev = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.q = [fenics.Function(V) for V in self.form_handler.control_spaces]
self.q_prev = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.hessian_actions = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.inactive_part = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.active_part = [
fenics.Function(V) for V in self.form_handler.control_spaces
]
self.controls = self.form_handler.controls
self.rtol = self.config.getfloat('StateSystem',
'picard_rtol',
fallback=1e-10)
self.atol = self.config.getfloat('StateSystem',
'picard_atol',
fallback=1e-12)
self.maxiter = self.config.getint('StateSystem',
'picard_iter',
fallback=50)
self.picard_verbose = self.config.getboolean('StateSystem',
'picard_verbose',
fallback=False)
self.no_sensitivity_solves = 0
self.state_ksps = [
PETSc.KSP().create() for i in range(self.form_handler.state_dim)
]
_setup_petsc_options(self.state_ksps,
self.form_handler.state_ksp_options)
self.adjoint_ksps = [
PETSc.KSP().create() for i in range(self.form_handler.state_dim)
]
_setup_petsc_options(self.adjoint_ksps,
self.form_handler.adjoint_ksp_options)
# Initialize the PETSc Krylov solver for the Riesz projection problems
self.ksps = [PETSc.KSP().create() for i in range(self.control_dim)]
option = [['ksp_type', 'cg'], ['pc_type', 'hypre'],
['pc_hypre_type', 'boomeramg'],
['pc_hypre_boomeramg_strong_threshold', 0.7],
['ksp_rtol', 1e-16], ['ksp_atol', 1e-50],
['ksp_max_it', 100]]
self.riesz_ksp_options = []
for i in range(self.control_dim):
self.riesz_ksp_options.append(option)
_setup_petsc_options(self.ksps, self.riesz_ksp_options)
for i, ksp in enumerate(self.ksps):
ksp.setOperators(self.form_handler.riesz_projection_matrices[i])
