def __init__(self, mesh, Vh_STATE, x0):
"""
Constructor.
INPUTS:
- mesh: the mesh
- Vh_STATE: the finite element space for the state variable
- x0: location at which we want to compute the jet-thickness
"""
Vh_help = dl.FunctionSpace(mesh, "CG", 1)
xfun = dl.interpolate(dl.Expression("x[0]", degree=1), Vh_help)
x_coord = xfun.vector().gather_on_zero()
mpi_comm = mesh.mpi_comm()
rank = dl.MPI.rank(mpi_comm)
nproc = dl.MPI.size(mpi_comm)
# round x0 so that it is aligned with the mesh
if nproc > 1:
from mpi4py import MPI
comm = MPI.COMM_WORLD
if rank == 0:
idx = (np.abs(x_coord - x0)).argmin()
self.x0 = x_coord[idx]
else:
self.x0 = None
self.x0 = comm.bcast(self.x0, root=0)
else:
idx = (np.abs(x_coord - x0)).argmin()
self.x0 = x_coord[idx]
line_segment = dl.AutoSubDomain(lambda x: dl.near(x[0], self.x0))
markers_f = dl.FacetFunction("size_t", mesh)
markers_f.set_all(0)
line_segment.mark(markers_f, 1)
dS = dl.dS[markers_f]
x_test = dl.TestFunctions(Vh_STATE)
u_test = x_test[0]
e1 = dl.Constant(("1.", "0."))
self.int_u = dl.assemble(dl.avg(dl.dot(u_test, e1)) * dS(1))
#self.u_cl = dl.assemble( dl.dot(u_test,e1)*dP(1) )
self.u_cl = dl.Function(Vh_STATE).vector()
ps = dl.PointSource(Vh_STATE.sub(0).sub(0), dl.Point(self.x0, 0.), 1.)
ps.apply(self.u_cl)
scaling = self.u_cl.sum()
if np.abs(scaling - 1.) > 1e-6:
print scaling
raise ValueError()
self.state = dl.Function(Vh_STATE).vector()
self.help = dl.Function(Vh_STATE).vector()
def fem_pts(conductivity, pos_list, save_dest, ele_list=None, sel_idx=None):
NPYSave = False
HDF5Save = False
if save_dest.find('.h5') > -1:
HDF5Save = True
dump_file = d.HDF5File(d.mpi_comm_world(), save_dest, 'w')
elif save_dest.find('.npy') > -1:
if not ele_list:
print('Expecting ele_list argument')
else:
NPYSave = True
else:
print('Only .h5 (entire mesh space), .npy (known ele_pos) supported')
print('Will not save anything this time')
if not sel_idx:
sel_idx = range(len(pos_list))
print('On this process no. pt. srcs = ', len(sel_idx))
mesh, subdomain, boundaries = meshes.load_meshes()
sigma = sigma_tensor(mesh, conductivity=conductivity)
print('Done loading meshes and conductivity')
V = d.FunctionSpace(mesh, "CG", 2)
v = d.TestFunction(V)
u = d.TrialFunction(V)
dx = d.Measure("dx")(subdomain_data=subdomain)
# ds = d.Measure("ds")(subdomain_data=boundaries)
a = d.inner(sigma * d.grad(u), d.grad(v)) * dx
L = d.Constant(0) * v * dx()
A = d.assemble(a)
# Surface of the grnd ele = 1030
bc = d.DirichletBC(V, d.Constant(0), boundaries, 1030)
for curr_idx in sel_idx:
solver = set_solver()
phi = d.Function(V)
x = phi.vector()
print('Started computing for,at: ', curr_idx, pos_list[curr_idx])
b = d.assemble(L)
bc.apply(A, b)
xx, yy, zz = pos_list[curr_idx]
point = d.Point(xx, yy, zz)
delta = d.PointSource(V, point, 1.)
delta.apply(b)
solver.solve(A, x, b)
# file = d.File("pots_anis.pvd")
# file << phi
if HDF5Save:
dump_file.write(x.array(), str(curr_idx))
dump_file.flush()
if NPYSave:
vals = extract_pots(phi, np.array(ele_list))
np.save(save_dest, vals)
print('Finished computing for :', curr_idx)
return
def onclick(event):
if print_point:
print('(', event.xdata, ',', event.ydata, ')')
delta_p = dl.PointSource(function_space_V, dl.Point(event.xdata, event.ydata), 1.0)
if event.button == 3:
point_source_dual_vector[:] = 0.
delta_p.apply(point_source_dual_vector)
Adelta.vector()[:] = apply_A(point_source_dual_vector)
plt.clf()
c = dl.plot(Adelta)
plt.colorbar(c)
plt.title('left click adds points, right click resets')
plt.draw()
def ptsrc(self,u,ord=1):
"""
Point source of (ord) order derivative of data-misfit function wrt. the solution u.
"""
assert ord in [1,2], 'Wrong order!'
u_vec = self._extr_soloc(u)
# define PointSource similar to boundary function, but PointSource is applied to (rhs) vector and is limited to scalar FunctionSpace
dfun_vec = u_vec
if ord==1:
dfun_vec -= self.obs
dfun_vec *= self.prec
dirac = [df.PointSource(self.pde.W.sub(0),df.Point(p),f) for (p,f) in zip(self.loc,dfun_vec)] # fails in 1.6.0 (mac app) possibly due to swig bug in numpy.i (already fixed in numpy 1.10.2) of the system numpy 1.8.0rc1
return dirac
def accountPointSource(self):
#-----------------------------------------------------------------------------------
componentIndex = {'real': 0, 'imag': 1}
for sourceComponent in ('real', 'imag'):
if self.sourceOpt[sourceComponent]['choice'] == 'pointSource':
pointSourceMag = self.sourceOpt[sourceComponent][
'pointSourceMag']
pointSourceLoc = self.sourceOpt[sourceComponent][
'pointSourceLoc']
for funSpaceComponent in ('real', 'imag'):
if sourceComponent == 'imag' and funSpaceComponent == 'imag':
pointSourceMag = -pointSourceMag
PS = df.PointSource(self.V.sub(componentIndex[funSpaceComponent]),\
df.Point(pointSourceLoc),pointSourceMag)
PS.apply(self.rhsVec)
return
# Define function space and basis functions
# This corresponds to Neumann boundary conditions zero, i.e. all outer boundaries are insulating.
L = df.Constant(0) * v * dx
# Define Dirichlet boundary conditions at left and right boundaries
bcs = [df.DirichletBC(V, 0.0, boundaries, 1)]
for t_idx in range(num_tsteps):
print("Time step {} of {}".format(t_idx, num_tsteps))
phi = df.Function(V)
A = df.assemble(a)
b = df.assemble(L)
[bc.apply(A, b) for bc in bcs]
# Adding point sources from neural simulation
for s_idx, s_pos in enumerate(source_pos):
point = df.Point(s_pos[0], s_pos[1], s_pos[2])
delta = df.PointSource(V, point, imem[s_idx, t_idx])
delta.apply(b)
df.solve(A, phi.vector(), b, 'cg', "ilu")
# df.File(join(out_folder, "phi_t_vec_{}.xml".format(t_idx))) << phi
np.save(join(out_folder, "phi_t_vec_{}.npy".format(t_idx)), phi.vector())
plot_FEM_results(phi, t_idx)
def run_model(function_space,
kappa,
forcing,
init_condition,
dt,
final_time,
boundary_conditions=None,
second_order_timestepping=False,
exact_sol=None,
velocity=None,
point_sources=None,
intermediate_times=None):
"""
Use implicit euler to solve transient advection diffusion equation
du/dt = grad (k* grad u) - vel*grad u + f
WARNINGarningW: when point sources solution changes significantly when mesh is varied
"""
mesh = function_space.mesh()
time_independent_boundaries = False
if boundary_conditions == None:
bndry_obj = dl.CompiledSubDomain("on_boundary")
boundary_conditions = [['dirichlet', bndry_obj, dl.Constant(0)]]
time_independent_boundaries = True
num_bndrys = len(boundary_conditions)
boundaries = mark_boundaries(mesh, boundary_conditions)
dirichlet_bcs = collect_dirichlet_boundaries(function_space,
boundary_conditions,
boundaries)
# To express integrals over the boundary parts using ds(i), we must first
# redefine the measure ds in terms of our boundary markers:
ds = dl.Measure('ds', domain=mesh, subdomain_data=boundaries)
dx = dl.Measure('dx', domain=mesh)
# Variational problem at each time
u = dl.TrialFunction(function_space)
v = dl.TestFunction(function_space)
# Previous solution
if hasattr(init_condition, 't'):
assert init_condition.t == 0
u_1 = dl.interpolate(init_condition, function_space)
if not second_order_timestepping:
theta = 1
else:
theta = 0.5
if hasattr(forcing, 't'):
forcing_1 = copy_expression(forcing)
else:
forcing_1 = forcing
def steady_state_form(u, v, f):
F = kappa * dl.inner(dl.grad(u), dl.grad(v)) * dx
F -= f * v * dx
if velocity is not None:
F += dl.dot(velocity, dl.grad(u)) * v * dx
return F
F = u*v*dx-u_1*v*dx + dt*theta*steady_state_form(u,v,forcing) + \
dt*(1.-theta)*steady_state_form(u_1,v,forcing_1)
a, L = dl.lhs(F), dl.rhs(F)
# a = u*v*dx + theta*dt*kappa*dl.inner(dl.grad(u), dl.grad(v))*dx
# L = (u_1 + dt*theta*forcing)*v*dx
# if velocity is not None:
# a += theta*dt*v*dl.dot(velocity,dl.grad(u))*dx
# if second_order_timestepping:
# L -= (1-theta)*dt*dl.inner(kappa*dl.grad(u_1), dl.grad(v))*dx
# L += (1-theta)*dt*forcing_1*v*dx
# if velocity is not None:
# L -= (1-theta)*dt*(v*dl.dot(velocity,dl.grad(u_1)))*dx
beta_1_list = []
alpha_1_list = []
for ii in range(num_bndrys):
if (boundary_conditions[ii][0] == 'robin'):
alpha = boundary_conditions[ii][3]
a += theta * dt * alpha * u * v * ds(ii)
if second_order_timestepping:
if hasattr(alpha, 't'):
alpha_1 = copy_expression(alpha)
alpha_1_list.append(alpha_1)
else:
alpha_1 = alpha
L -= (1 - theta) * dt * alpha_1 * u_1 * v * ds(ii)
if ((boundary_conditions[ii][0] == 'robin')
or (boundary_conditions[ii][0] == 'neumann')):
beta = boundary_conditions[ii][2]
L -= theta * dt * beta * v * ds(ii)
if second_order_timestepping:
if hasattr(beta, 't'):
beta_1 = copy_expression(beta)
beta_1_list.append(beta_1)
else:
# boundary condition is constant in time
beta_1 = beta
L -= (1 - theta) * dt * beta_1 * v * ds(ii)
if time_independent_boundaries:
# TODO this can be used if dirichlet and robin conditions are not
# time dependent.
A = dl.assemble(a)
for bc in dirichlet_bcs:
bc.apply(A)
solver = dl.LUSolver(A)
#solver.parameters["reuse_factorization"] = True
else:
solver = None
u_2 = dl.Function(function_space)
u_2.assign(u_1)
t = 0.0
dt_tol = 1e-12
n_time_steps = 0
if intermediate_times is not None:
intermediate_u = []
intermediate_cnt = 0
# assert in chronological order
assert np.allclose(intermediate_times, np.array(intermediate_times))
assert np.all(intermediate_times < final_time)
while t < final_time - dt_tol:
# Update current time
t += dt
forcing.t = t
forcing_1.t = t - dt
# set current time for time varying boundary conditions
for ii in range(num_bndrys):
if hasattr(boundary_conditions[ii][2], 't'):
boundary_conditions[ii][2].t = t
# set previous time for time varying boundary conditions when
# using second order timestepping. lists will be empty if using
# first order timestepping
for jj in range(len(beta_1_list)):
beta_1_list[jj].t = t - dt
for jj in range(len(alpha_1_list)):
alpha_1_list[jj].t = t - dt
#A, b = dl.assemble_system(a, L, dirichlet_bcs)
#for bc in dirichlet_bcs:
# bc.apply(A,b)
if boundary_conditions is not None:
A = dl.assemble(a)
for bc in dirichlet_bcs:
bc.apply(A)
b = dl.assemble(L)
for bc in dirichlet_bcs:
bc.apply(b)
if point_sources is not None:
ps_list = []
for ii in range(len(point_sources)):
point, expr = point_sources[ii]
ps_list.append((dl.Point(point[0], point[1]), expr(t)))
ps = dl.PointSource(function_space, ps_list)
ps.apply(b)
if solver is None:
dl.solve(A, u_2.vector(), b)
else:
solver.solve(u_2.vector(), b)
#print ("t =", t, "end t=", final_time)
# Update previous solution
u_1.assign(u_2)
# import matplotlib.pyplot as plt
# plt.subplot(131)
# pp=dl.plot(u_1)
# plt.subplot(132)
# dl.plot(forcing,mesh=mesh)
# plt.subplot(133)
# dl.plot(forcing_1,mesh=mesh)
# plt.colorbar(pp)
# plt.show()
# compute error
if exact_sol is not None:
exact_sol.t = t
error = dl.errornorm(exact_sol, u_2)
print('t = %.2f: error = %.3g' % (t, error))
#dl.plot(exact_sol,mesh=mesh)
#plt.show()
t = min(t, final_time)
if (intermediate_times is not None
and intermediate_cnt < intermediate_times.shape[0]
and t >= intermediate_times[intermediate_cnt]):
#save solution closest to intermediate time
u_t = dl.Function(function_space)
u_t.assign(u_2)
intermediate_u.append(u_t)
intermediate_cnt += 1
n_time_steps += 1
#print ("t =", t, "end t=", final_time,"# time steps", n_time_steps)
if intermediate_times is None:
return u_2
else:
return intermediate_u + [u_2]
def simulate_FEM():
import dolfin as df
df.parameters['allow_extrapolation'] = False
# Define mesh
mesh = df.Mesh(join(mesh_folder, "{}.xml".format(mesh_name)))
subdomains = df.MeshFunction("size_t", mesh, join(mesh_folder,
"{}_physical_region.xml".format(mesh_name)))
boundaries = df.MeshFunction("size_t", mesh, join(mesh_folder,
"{}_facet_region.xml".format(mesh_name)))
print("Number of cells in mesh: ", mesh.num_cells())
np.save(join(out_folder, "mesh_coordinates.npy"), mesh.coordinates())
sigma_vec = df.Constant(sigma)
V = df.FunctionSpace(mesh, "CG", 2)
v = df.TestFunction(V)
u = df.TrialFunction(V)
ds = df.Measure("ds", domain=mesh, subdomain_data=boundaries)
dx = df.Measure("dx", domain=mesh, subdomain_data=subdomains)
a = df.inner(sigma_vec * df.grad(u), df.grad(v)) * dx(1)
# This corresponds to Neumann boundary conditions zero, i.e.
# all outer boundaries are insulating.
L = df.Constant(0) * v * dx
# Define Dirichlet boundary conditions outer cylinder boundaries (ground)
bcs = [df.DirichletBC(V, 0.0, boundaries, 1)]
for t_idx in range(num_tsteps):
f_name = join(out_folder, "phi_xz_t_vec_{}.npy".format(t_idx))
# if os.path.isfile(f_name):
# print("skipping ", f_name)
# continue
print("Time step {} of {}".format(t_idx, num_tsteps))
phi = df.Function(V)
A = df.assemble(a)
b = df.assemble(L)
[bc.apply(A, b) for bc in bcs]
# Adding point sources from neural simulation
for s_idx, s_pos in enumerate(source_pos):
point = df.Point(s_pos[0], s_pos[1], s_pos[2])
delta = df.PointSource(V, point, imem[s_idx, t_idx])
delta.apply(b)
df.solve(A, phi.vector(), b, 'cg', "ilu")
# df.File(join(out_folder, "phi_t_vec_{}.xml".format(t_idx))) << phi
# np.save(join(out_folder, "phi_t_vec_{}.npy".format(t_idx)), phi.vector())
plot_and_save_simulation_results(phi, t_idx)
def run(self, species="Oxygen", increment=0):
# Specify the test/trial function space
df.set_log_active(False)
V = df.FunctionSpace(self.mesh, "Lagrange", 1)
# Specify the boundary conditions, Dirichlet on all domain faces
def u0_boundary(x, on_boundary):
return on_boundary
# Define the problem
u = df.TrialFunction(V)
v = df.TestFunction(V)
if species == "Oxygen":
bc = df.DirichletBC(V, df.Constant(1), u0_boundary)
f = df.Constant(0.0)
a = self.Dc * df.inner(df.nabla_grad(u), df.nabla_grad(v)) * df.dx
L = f * v * df.dx
elif species == "Factor":
bc = df.DirichletBC(V, df.Constant(0), u0_boundary)
f = df.Constant(0)
a = (self.decayRate * u * v + self.Dv *
df.inner(df.nabla_grad(u), df.nabla_grad(v))) * df.dx
L = f * v * df.dx
# Assemble the system
A, b = df.assemble_system(a, L, bc)
if species == "Oxygen":
# Add vessel source terms
vesselSources = self.sources[1]
for eachSource in vesselSources:
location = [
point - self.spacing / 2.0 for point in eachSource[0]
]
if self.extents[2] > 1:
delta = df.PointSource(
V, df.Point(location[0], location[1], location[2]),
self.permeability * eachSource[1])
else:
delta = df.PointSource(V, df.Point(location[0],
location[1]),
self.permeability * eachSource[1])
try:
delta.apply(b)
except:
pass
# Add cell sink terms
cellSources = self.sources[0]
for eachSource in cellSources:
location = [
point - self.spacing / 2.0 for point in eachSource[0]
]
if self.extents[2] > 1:
delta = df.PointSource(
V, df.Point(location[0], location[1], location[2]),
-self.consumptionRate * eachSource[1])
else:
delta = df.PointSource(
V, df.Point(location[0], location[1]),
-self.consumptionRate * eachSource[1])
try:
delta.apply(b)
except:
pass
elif species == "Factor":
# Add cell source terms
cellSources = self.sources[0]
for eachSource in cellSources:
location = [
point - self.spacing / 2.0 for point in eachSource[0]
]
if self.extents[2] > 1:
delta = df.PointSource(
V, df.Point(location[0], location[1], location[2]),
self.factorSensitvity * eachSource[1])
else:
delta = df.PointSource(
V, df.Point(location[0], location[1]),
self.factorSensitvity * eachSource[1])
try:
delta.apply(b)
except:
pass
# Set up solution vector
u = df.Function(V)
U = u.vector()
# Set up and run solver
solver = df.KrylovSolver("cg", "ilu")
solver.solve(A, U, b)
self.result = []
for eachEntry in self.sources[0]:
location = [point - self.spacing / 2.0 for point in eachEntry[0]]
if self.extents[2] <= 1:
location = location[:2]
try:
result = u(location)
except:
if species == "Oxygen":
result = 1.0
else:
result = 0.0
self.result.append(result)
self.write_output(species, increment)
return self.result
def solve_adjoint(evo, control, ps_magnitude, target_point, a, V, j):
'''Calculates the solution to the adjoint problem.
The solution to the adjoint equation is calculated using the explicitly
given evolution (solution to the forward problem) and the control.
For better understanding of the indeces see docs/indexing-diagram.txt.
Parameters: OUTDATED
V: dolfin.FunctionSpace
The FEM space of the problem being solved.
evo: ndarray
The coefficients of the solution to the corresponding forward
problem in the basis of the space V (see solve_forward).
control: ndarray
The laser power profile.
beta_welding
target_point: dolfin.Point
threshold_temp
penalty_term_combined
Returns:
evo_adj: ndarray
The coefficients of the calculated adjoint solution in the basis of
the space V.
'''
# initialize state functions
theta_km1 = dolfin.Function(V) # stands for theta[k-1]
theta_k = dolfin.Function(V) # stands for theta[k]
theta_kp1 = dolfin.Function(V) # stands for theta[k+1]
# initialize adjoint state functions
p_km1 = dolfin.Function(V) # stands for p[k-1]
p_k = dolfin.Function(V) # stands for p[k]
# FEM equation setup
p = dolfin.TrialFunction(V) # stands for unknown p[k-1]
v = dolfin.TestFunction(V)
Nt = len(control)
evo_adj = np.zeros((Nt+1, len(V.dofmap().dofs())))
# preparing for the first iteration
theta_k.vector().set_local(evo[Nt])
# solve backward, i.e. p_k -> p = p_km1, k = Nt, Nt-1, Nt-2, ..., 1
for k in range(Nt, 0, -1):
theta_km1.vector().set_local(evo[k-1])
F = a(theta_km1, theta_k, p, control[k-1])\
+ j(k-1, theta_km1, theta_k)
if k < Nt:
F += a(theta_k, theta_kp1, p_k, control[k])\
+ j(k, theta_k, theta_kp1)
dF = dolfin.derivative(F, theta_k, v)
# sometimes rhs(dF) is void which leads to a ValueError
try:
A, b = dolfin.assemble_system(dolfin.lhs(dF), dolfin.rhs(dF))
except ValueError:
A, b = dolfin.assemble_system(dolfin.lhs(dF), Constant(0)*v*dx)
# apply welding penalty as a point source
point_source = dolfin.PointSource(V, target_point, ps_magnitude[k-1])
point_source.apply(b)
dolfin.solve(A, p_km1.vector(), b)
evo_adj[k-1] = p_km1.vector().get_local()
# preparing for the next iteration
p_k.assign(p_km1)
theta_kp1.assign(theta_k)
theta_k.assign(theta_km1)
return evo_adj