def get_poisson_steps(pts, cells, tol):
# Still can't initialize a mesh from points, cells
filename = "mesh.xdmf"
if cells.shape[1] == 3:
meshio.write_points_cells(filename, pts, {"triangle": cells})
else:
assert cells.shape[1] == 4
meshio.write_points_cells(filename, pts, {"tetra": cells})
mesh = Mesh()
with XDMFFile(filename) as f:
f.read(mesh)
os.remove(filename)
os.remove("mesh.h5")
# build Laplace matrix with Dirichlet boundary using dolfin
V = FunctionSpace(mesh, "Lagrange", 1)
u = TrialFunction(V)
v = TestFunction(V)
a = inner(grad(u), grad(v)) * dx
u0 = Constant(0.0)
bc = DirichletBC(V, u0, "on_boundary")
f = Constant(1.0)
L = f * v * dx
A = assemble(a)
b = assemble(L)
bc.apply(A, b)
# solve(A, x, b, "cg")
solver = KrylovSolver("cg", "none")
solver.parameters["absolute_tolerance"] = 0.0
solver.parameters["relative_tolerance"] = tol
x = Function(V)
x_vec = x.vector()
num_steps = solver.solve(A, x_vec, b)
# # convert to scipy matrix
# A = as_backend_type(A).mat()
# ai, aj, av = A.getValuesCSR()
# A = scipy.sparse.csr_matrix(
# (av, aj, ai), shape=(A.getLocalSize()[0], A.getSize()[1])
# )
# # ev = eigvals(A.todense())
# ev_max = scipy.sparse.linalg.eigs(A, k=1, which="LM")[0][0]
# assert numpy.abs(ev_max.imag) < 1.0e-15
# ev_max = ev_max.real
# ev_min = scipy.sparse.linalg.eigs(A, k=1, which="SM")[0][0]
# assert numpy.abs(ev_min.imag) < 1.0e-15
# ev_min = ev_min.real
# cond = ev_max / ev_min
# solve poisson system, count num steps
# b = numpy.ones(A.shape[0])
# out = pykry.gmres(A, b)
# num_steps = len(out.resnorms)
return num_steps
def cyclic3D(u):
""" Symmetrize with respect to (xyz) cycle. """
try:
nrm = np.linalg.norm(u.vector())
V = u.function_space()
assert V.mesh().topology().dim() == 3
mesh1 = Mesh(V.mesh())
mesh1.coordinates()[:, :] = mesh1.coordinates()[:, [1, 2, 0]]
W1 = FunctionSpace(mesh1, 'CG', 1)
# testing if symmetric
bc = DirichletBC(V, 1, DomainBoundary())
test = Function(V)
bc.apply(test.vector())
test = interpolate(Function(W1, test.vector()), V)
assert max(test.vector()) - min(test.vector()) < 1.1
v1 = interpolate(Function(W1, u.vector()), V)
mesh2 = Mesh(mesh1)
mesh2.coordinates()[:, :] = mesh2.coordinates()[:, [1, 2, 0]]
W2 = FunctionSpace(mesh2, 'CG', 1)
v2 = interpolate(Function(W2, u.vector()), V)
pr = project(u+v1+v2)
assert np.linalg.norm(pr.vector())/nrm > 0.01
return pr
except:
print "Cyclic symmetrization failed!"
return u
class StreamFunction(Field):
def before_first_compute(self, get):
u = get("Velocity")
assert len(u) == 2, "Can only compute stream function for 2D problems"
V = u.function_space()
spaces = SpacePool(V.mesh())
degree = V.ufl_element().degree()
V = spaces.get_space(degree, 0)
psi = TrialFunction(V)
self.q = TestFunction(V)
a = dot(grad(psi), grad(self.q)) * dx()
self.bc = DirichletBC(V, Constant(0), DomainBoundary())
self.A = assemble(a)
self.L = Vector()
self.bc.apply(self.A)
self.solver = KrylovSolver(self.A, "cg")
self.psi = Function(V)
def compute(self, get):
u = get("Velocity")
assemble(dot(u[1].dx(0) - u[0].dx(1), self.q) * dx(), tensor=self.L)
self.bc.apply(self.L)
self.solver.solve(self.psi.vector(), self.L)
#solve(self.A, self.psi.vector(), self.L)
return self.psi
class StreamFunction(Field):
def before_first_compute(self, get):
u = get("Velocity")
assert len(u) == 2, "Can only compute stream function for 2D problems"
V = u.function_space()
spaces = SpacePool(V.mesh())
degree = V.ufl_element().degree()
V = spaces.get_space(degree, 0)
psi = TrialFunction(V)
self.q = TestFunction(V)
a = dot(grad(psi), grad(self.q))*dx()
self.bc = DirichletBC(V, Constant(0), DomainBoundary())
self.A = assemble(a)
self.L = Vector()
self.bc.apply(self.A)
self.solver = KrylovSolver(self.A, "cg")
self.psi = Function(V)
def compute(self, get):
u = get("Velocity")
assemble(dot(u[1].dx(0)-u[0].dx(1), self.q)*dx(), tensor=self.L)
self.bc.apply(self.L)
self.solver.solve(self.psi.vector(), self.L)
#solve(self.A, self.psi.vector(), self.L)
return self.psi
def cyclic3D(u):
""" Symmetrize with respect to (xyz) cycle. """
try:
nrm = np.linalg.norm(u.vector())
V = u.function_space()
assert V.mesh().topology().dim() == 3
mesh1 = Mesh(V.mesh())
mesh1.coordinates()[:, :] = mesh1.coordinates()[:, [1, 2, 0]]
W1 = FunctionSpace(mesh1, 'CG', 1)
# testing if symmetric
bc = DirichletBC(V, 1, DomainBoundary())
test = Function(V)
bc.apply(test.vector())
test = interpolate(Function(W1, test.vector()), V)
assert max(test.vector()) - min(test.vector()) < 1.1
v1 = interpolate(Function(W1, u.vector()), V)
mesh2 = Mesh(mesh1)
mesh2.coordinates()[:, :] = mesh2.coordinates()[:, [1, 2, 0]]
W2 = FunctionSpace(mesh2, 'CG', 1)
v2 = interpolate(Function(W2, u.vector()), V)
pr = project(u + v1 + v2)
assert np.linalg.norm(pr.vector()) / nrm > 0.01
return pr
except:
print "Cyclic symmetrization failed!"
return u
def assert_solves(
mesh: Mesh,
diffusion: Coefficient,
convection: Optional[Coefficient],
reaction: Optional[Coefficient],
source: Coefficient,
exact: Coefficient,
l2_tol: Optional[float] = 1.0e-8,
h1_tol: Optional[float] = 1.0e-6,
):
eafe_matrix = eafe_assemble(mesh, diffusion, convection, reaction)
pw_linears = FunctionSpace(mesh, "Lagrange", 1)
test_function = TestFunction(pw_linears)
rhs_vector = assemble(source * test_function * dx)
bc = DirichletBC(pw_linears, exact, lambda _, on_bndry: on_bndry)
bc.apply(eafe_matrix, rhs_vector)
solution = Function(pw_linears)
solver = LUSolver(eafe_matrix, "default")
solver.parameters["symmetric"] = False
solver.solve(solution.vector(), rhs_vector)
l2_err: float = errornorm(exact, solution, "l2", 3)
assert l2_err <= l2_tol, f"L2 error too large: {l2_err} > {l2_tol}"
h1_err: float = errornorm(exact, solution, "H1", 3)
assert h1_err <= h1_tol, f"H1 error too large: {h1_err} > {h1_tol}"
class Heat(object):
'''
u' = \\Delta u + f
'''
def __init__(self, V):
self.sol = Expression(sympy.printing.ccode(solution),
degree=MAX_DEGREE,
t=0.0,
cell=triangle)
self.V = V
u = TrialFunction(V)
v = TestFunction(V)
self.M = assemble(u * v * dx)
n = FacetNormal(self.V.mesh())
self.A = assemble(-inner(kappa * grad(u), grad(v /
(rho * cp))) * dx +
inner(kappa * grad(u), n) * v / (rho * cp) * ds)
self.bcs = DirichletBC(self.V, self.sol, 'on_boundary')
return
def eval_alpha_M_beta_F(self, alpha, beta, u, t):
# Evaluate alpha * M * u + beta * F(u, t).
uvec = u.vector()
v = TestFunction(self.V)
f.t = t
b = assemble(f * v / (rho * cp) * dx)
out = Function(self.V)
out.vector()[:] = \
alpha * (self.M * uvec) + beta * (self.A * uvec + b)
return out
def solve_alpha_M_beta_F(self, alpha, beta, b, t):
# Solve alpha * M * u + beta * F(u, t) = b for u.
A = alpha * self.M + beta * self.A
f.t = t
v = TestFunction(self.V)
b2 = f / (rho * cp)
rhs = assemble((b - beta * b2) * v * dx)
self.bcs.apply(A, rhs)
solver = \
KrylovSolver('gmres', 'ilu') if alpha < 0.0 or beta > 0.0 \
else KrylovSolver('cg', 'amg')
solver.parameters['relative_tolerance'] = 1.0e-13
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['maximum_iterations'] = 100
solver.parameters['monitor_convergence'] = False
solver.set_operator(A)
u = Function(self.V)
solver.solve(u.vector(), rhs)
return u
class Bratu(object):
def __init__(self):
self.mesh = UnitSquareMesh(40, 40, "left/right")
self.V = FunctionSpace(self.mesh, "Lagrange", 1)
self.bc = DirichletBC(self.V, 0.0, "on_boundary")
u = TrialFunction(self.V)
v = TestFunction(self.V)
self.a = assemble(dot(grad(u), grad(v)) * dx)
self.m = assemble(u * v * dx)
def inner(self, a, b):
return a.inner(self.m * b)
def norm2_r(self, a):
return a.inner(a)
def f(self, u, lmbda):
v = TestFunction(self.V)
ufun = Function(self.V)
ufun.vector()[:] = u
out = self.a * u - lmbda * assemble(exp(ufun) * v * dx)
DirichletBC(self.V, ufun, "on_boundary").apply(out)
return out
def df_dlmbda(self, u, lmbda):
v = TestFunction(self.V)
ufun = Function(self.V)
ufun.vector()[:] = u
out = -assemble(exp(ufun) * v * dx)
self.bc.apply(out)
return out
def jacobian_solver(self, u, lmbda, rhs):
t = TrialFunction(self.V)
v = TestFunction(self.V)
# from dolfin import Constant
# a = assemble(
# dot(grad(t), grad(v)) * dx - Constant(lmbda) * exp(u) * t * v * dx
# )
ufun = Function(self.V)
ufun.vector()[:] = u
a = self.a - lmbda * assemble(exp(ufun) * t * v * dx)
self.bc.apply(a)
x = Function(self.V)
# solve(a, x.vector(), rhs, "gmres", "ilu")
solve(a, x.vector(), rhs)
return x.vector()
def symmetrize(u, d, sym):
""" Symmetrize function u. """
if len(d) == 3:
# three dimensions -> cycle XYZ
return cyclic3D(u)
elif len(d) >= 4:
# four dimensions -> rotations in 2D
return rotational(u, d[-1])
nrm = np.linalg.norm(u.vector())
V = u.function_space()
mesh = Mesh(V.mesh())
# test if domain is symmetric using function equal 0 inside, 1 on boundary
# extrapolation will force large values if not symmetric since the flipped
# domain is different
bc = DirichletBC(V, 1, DomainBoundary())
test = Function(V)
bc.apply(test.vector())
if len(d) == 2:
# two dimensions given: swap dimensions
mesh.coordinates()[:, d] = mesh.coordinates()[:, d[::-1]]
else:
# one dimension given: reflect
mesh.coordinates()[:, d[0]] *= -1
# FIXME functionspace takes a long time to construct, maybe copy?
W = FunctionSpace(mesh, 'CG', 1)
try:
# testing
test = interpolate(Function(W, test.vector()), V)
# max-min should be around 1 if domain was symmetric
# may be slightly above due to boundary approximation
assert max(test.vector()) - min(test.vector()) < 1.1
v = interpolate(Function(W, u.vector()), V)
if sym:
# symmetric
pr = project(u+v)
else:
# antisymmetric
pr = project(u-v)
# small solution norm most likely means that symmetrization gives
# trivial function
assert np.linalg.norm(pr.vector())/nrm > 0.01
return pr
except:
# symmetrization failed for some reason
print "Symmetrization " + str(d) + " failed!"
return u
def symmetrize(u, d, sym):
""" Symmetrize function u. """
if len(d) == 3:
# three dimensions -> cycle XYZ
return cyclic3D(u)
elif len(d) >= 4:
# four dimensions -> rotations in 2D
return rotational(u, d[-1])
nrm = np.linalg.norm(u.vector())
V = u.function_space()
mesh = Mesh(V.mesh())
# test if domain is symmetric using function equal 0 inside, 1 on boundary
# extrapolation will force large values if not symmetric since the flipped
# domain is different
bc = DirichletBC(V, 1, DomainBoundary())
test = Function(V)
bc.apply(test.vector())
if len(d) == 2:
# two dimensions given: swap dimensions
mesh.coordinates()[:, d] = mesh.coordinates()[:, d[::-1]]
else:
# one dimension given: reflect
mesh.coordinates()[:, d[0]] *= -1
# FIXME functionspace takes a long time to construct, maybe copy?
W = FunctionSpace(mesh, 'CG', 1)
try:
# testing
test = interpolate(Function(W, test.vector()), V)
# max-min should be around 1 if domain was symmetric
# may be slightly above due to boundary approximation
assert max(test.vector()) - min(test.vector()) < 1.1
v = interpolate(Function(W, u.vector()), V)
if sym:
# symmetric
pr = project(u + v)
else:
# antisymmetric
pr = project(u - v)
# small solution norm most likely means that symmetrization gives
# trivial function
assert np.linalg.norm(pr.vector()) / nrm > 0.01
return pr
except:
# symmetrization failed for some reason
print "Symmetrization " + str(d) + " failed!"
return u
def solve(self, L, a, plate_potential):
A = assemble(a)
b = assemble(L)
# print(' assembled')
dirichlet_bc = DirichletBC(self.V, plate_potential,
(lambda x, on_boundary: on_boundary and x[2]
< self.GROUNDED_PLATE_AT))
dirichlet_bc.apply(A, b)
# print(' modified')
f = self.function()
self.solver.solve(A, f.vector(), b)
# print(' solved')
return f
def dolfin_comparasion(Nx, Ny, f):
from dolfin import (UnitSquareMesh, FunctionSpace, TrialFunction,
TestFunction, assemble, inner, grad, dx,
SpatialCoordinate, DirichletBC, Function, solve, pi,
CellType, sin, cos)
mesh = UnitSquareMesh.create(Nx, Ny, CellType.Type.quadrilateral)
V = FunctionSpace(mesh, "CG", 1)
u, v = TrialFunction(V), TestFunction(V)
a = inner(grad(u), grad(v)) * dx
A_dol = assemble(a)
x, y = SpatialCoordinate(mesh)
f = eval(f)
l_ = inner(f, v) * dx
B_dol = assemble(l_)
bc = DirichletBC(V, 0, "on_boundary")
bc.apply(A_dol, B_dol)
u_h = Function(V)
solve(A_dol, u_h.vector(), B_dol)
return u_h, assemble(u_h * dx)
def initial_solution(self, mode_number):
main_boundary = self.outsides
left_boundary = self.left_outsides
right_boundary = self.right_outsides
KK = self.K
bc = DirichletBC(self.V, Constant(0.), main_boundary)
bc.apply(KK)
solver = standard_solver(KK, self.M)
solver.solve(10)
KK1 = self.K1
bc1 = DirichletBC(self.V1, Constant(0.), left_boundary)
bc1.apply(KK1)
solver1 = standard_solver(KK1, self.M1)
solver1.solve(10)
KK2 = self.K2
bc2 = DirichletBC(self.V2, Constant(0.), right_boundary)
bc2.apply(KK2)
solver2 = standard_solver(KK2, self.M2)
solver2.solve(10)
r, _, rx, _ = solver.get_eigenpair(mode_number)
r1, _, rx1, _ = solver1.get_eigenpair(mode_number)
r2, _, rx2, _ = solver2.get_eigenpair(mode_number)
freqs = [r, r1, r2]
vecs = [rx, rx1, rx2]
return freqs, vecs
def _modify_linear_equation(self, x, y, z):
dx_src = f'(x[0] - {x})'
dy_src = f'(x[1] - {y})'
dz_src = f'(x[2] - {z})'
r_src2 = f'({dx_src} * {dx_src} + {dy_src} * {dy_src} + {dz_src} * {dz_src})'
r_src = f'sqrt({r_src2})'
conductivity = self.base_conductivity(x, y, z)
minus_potential_exp = Expression(f'''
{-0.25 / np.pi / conductivity}
/ {r_src}
''',
degree=self.degree,
domain=self._fm._mesh)
dirichlet_bc = DirichletBC(self._fm.function_space,
minus_potential_exp,
(lambda x, on_boundary: on_boundary and
x[2] < self.grounded_plate_edge_z))
logger.debug('Applying the "plate" Dirichlet BC')
dirichlet_bc.apply(self._terms_with_unknown, self._known_terms)
logger.debug('Done.')
class Heat(object):
'''
u' = \\Delta u + f
'''
def __init__(self, V):
self.V = V
u = TrialFunction(V)
v = TestFunction(V)
self.M = assemble(u * v * dx)
self.A = assemble(-dot(grad(u), grad(v)) * dx)
self.b = assemble(1.0 * v * dx)
self.bcs = DirichletBC(self.V, 0.0, 'on_boundary')
return
# pylint: disable=unused-argument
def eval_alpha_M_beta_F(self, alpha, beta, u, t):
# Evaluate alpha * M * u + beta * F(u, t).
uvec = u.vector()
return alpha * (self.M * uvec) + beta * (self.A * uvec + self.b)
def solve_alpha_M_beta_F(self, alpha, beta, b, t):
# Solve alpha * M * u + beta * F(u, t) = b for u.
A = alpha * self.M + beta * self.A
rhs = b - beta * self.b
self.bcs.apply(A, rhs)
solver = KrylovSolver('gmres', 'ilu')
solver.parameters['relative_tolerance'] = 1.0e-13
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['maximum_iterations'] = 100
solver.parameters['monitor_convergence'] = True
solver.set_operator(A)
u = Function(self.V)
solver.solve(u.vector(), rhs)
return u
def solve(self):
""" Find eigenvalues for transformed mesh. """
self.progress("Building mesh.")
# build transformed mesh
mesh = self.refineMesh()
# dim = mesh.topology().dim()
if self.bcLast:
mesh = transform_mesh(mesh, self.transformList)
Robin, Steklov, shift, bcs = get_bc_parts(mesh, self.bcList)
else:
Robin, Steklov, shift, bcs = get_bc_parts(mesh, self.bcList)
mesh = transform_mesh(mesh, self.transformList)
# boundary conditions computed on non-transformed mesh
# copy the values to transformed mesh
fun = FacetFunction("size_t", mesh, shift)
fun.array()[:] = bcs.array()[:]
bcs = fun
ds = Measure('ds', domain=mesh, subdomain_data=bcs)
V = FunctionSpace(mesh, self.method, self.deg)
u = TrialFunction(V)
v = TestFunction(V)
self.progress("Assembling matrices.")
wTop = Expression(self.wTop, degree=self.deg)
wBottom = Expression(self.wBottom, degree=self.deg)
#
# build stiffness matrix form
#
s = dot(grad(u), grad(v))*wTop*dx
# add Robin parts
for bc in Robin:
s += Constant(bc.parValue)*u*v*wTop*ds(bc.value+shift)
#
# build mass matrix form
#
if len(Steklov) > 0:
m = 0
for bc in Steklov:
m += Constant(bc.parValue)*u*v*wBottom*ds(bc.value+shift)
else:
m = u*v*wBottom*dx
# assemble
# if USE_EIGEN:
# S, M = EigenMatrix(), EigenMatrix()
# tempv = EigenVector()
# else:
S, M = PETScMatrix(), PETScMatrix()
# tempv = PETScVector()
if not np.any(bcs.array() == shift+1):
# no Dirichlet parts
assemble(s, tensor=S)
assemble(m, tensor=M)
else:
#
# with EIGEN we could
# apply Dirichlet condition symmetrically
# completely remove rows and columns
#
# Dirichlet parts are marked with shift+1
#
# temp = Constant(0)*v*dx
bc = DirichletBC(V, Constant(0.0), bcs, shift+1)
# assemble_system(s, temp, bc, A_tensor=S, b_tensor=tempv)
# assemble_system(m, temp, bc, A_tensor=M, b_tensor=tempv)
assemble(s, tensor=S)
bc.apply(S)
assemble(m, tensor=M)
# bc.zero(M)
# if USE_EIGEN:
# M = M.sparray()
# M.eliminate_zeros()
# print M.shape
# indices = M.indptr[:-1] - M.indptr[1:] < 0
# M = M[indices, :].tocsc()[:, indices]
# S = S.sparray()[indices, :].tocsc()[:, indices]
# print M.shape
#
# solve the eigenvalue problem
#
self.progress("Solving eigenvalue problem.")
eigensolver = SLEPcEigenSolver(S, M)
eigensolver.parameters["problem_type"] = "gen_hermitian"
eigensolver.parameters["solver"] = "krylov-schur"
if self.target is not None:
eigensolver.parameters["spectrum"] = "target real"
eigensolver.parameters["spectral_shift"] = self.target
else:
eigensolver.parameters["spectrum"] = "smallest magnitude"
eigensolver.parameters["spectral_shift"] = -0.01
eigensolver.parameters["spectral_transform"] = "shift-and-invert"
eigensolver.solve(self.number)
self.progress("Generating eigenfunctions.")
if eigensolver.get_number_converged() == 0:
return None
eigf = []
eigv = []
if self.deg > 1:
mesh = refine(mesh)
W = FunctionSpace(mesh, 'CG', 1)
for i in range(eigensolver.get_number_converged()):
pair = eigensolver.get_eigenpair(i)[::2]
eigv.append(pair[0])
u = Function(V)
u.vector()[:] = pair[1]
eigf.append(interpolate(u, W))
return eigv, eigf
def schwarz_algorithm(self, max_iterations, mode_number):
left_robin = self.left_robin
left_dirichlet = self.left_outsides
right_robin = self.right_robin
right_dirichlet = self.right_outsides
self.build_transfer_matrices()
B1 = self.B1
BO1 = self.BO1
B2 = self.B2
BO2 = self.BO2
V1 = self.V1
V2 = self.V2
VO = self.VO
freqs, vecs = self.initial_solution(mode_number)
r1 = freqs[1]
r2 = freqs[2]
M1 = self.M1
M2 = self.M2
rx1 = vecs[1]
rx2 = vecs[2]
if abs(rx1.max()) < abs(rx1.min()):
rx1 = self.adjustment * rx1 / (rx1.min())
else:
rx1 = self.adjustment * rx1 / (rx1.max())
if abs(rx2.max()) < abs(rx2.min()):
rx2 = self.adjustment * rx2 / (rx2.min())
else:
rx2 = self.adjustment * rx2 / (rx2.max())
iter = 0
L2_error = np.zeros((max_iterations + 1, 1))
H1_error = np.zeros((max_iterations + 1, 1))
SH_error = np.zeros((max_iterations + 1, 1))
r_left = np.zeros((max_iterations + 1, 1))
r_right = np.zeros((max_iterations + 1, 1))
u1 = Function(self.V1)
u1.vector()[:] = rx1
u2 = Function(self.V2)
u2.vector()[:] = rx2
L2, H1 = self.overlap_error_norms(u1, u2)
L2_error[iter] = L2
H1_error[iter] = H1
SH_error[iter] = H1 - L2
r_left[iter] = r1
r_right[iter] = r2
while (iter < max_iterations):
#Step 1: do f1 = du/dn2, g1 = u2
uu2 = Function(V2)
if (np.absolute(rx2.max()) < np.absolute(rx2.min())):
rx2 = self.adjustment * rx2 / (rx2.min())
uu2.vector()[:] = rx2
else:
rx2 = self.adjustment * rx2 / (rx2.max())
uu2.vector()[:] = rx2
uu21 = Function(VO)
uu21.vector()[:] = B2 * uu2.vector()
g1 = Function(V1)
g1.vector()[:] = BO1 * uu21.vector()
du2 = Function(V2)
n2 = FacetNormal(self.mesh2)
du2 = assemble(inner(grad(uu2), n2) * ds)
du2 = project(du2, V2)
du2O = Function(VO)
du2O.vector()[:] = B2 * du2.vector()
f1 = Function(V1)
f1.vector()[:] = BO1 * du2O.vector()
#Step 2: Solve for u1
boundary = left_robin
K1, _ = self.assemble_lui_stiffness(g=g1,
f=f1,
mesh=self.mesh1,
robin_boundary=boundary)
bc1 = DirichletBC(V1, Constant(0.), left_dirichlet)
bc1.apply(K1)
solver1 = standard_solver(K1, M1)
solver1.solve(10)
r1, _, rx1, _ = solver1.get_eigenpair(mode_number)
#Step 3: do f1 = du1/dn1, g2 = u1
uu1 = Function(V1)
if (np.absolute(rx1.max()) < np.absolute(rx1.min())):
uu1.vector()[:] = self.adjustment * rx1 / (rx1.min())
else:
uu1.vector()[:] = self.adjustment * rx1 / (rx1.max())
uu12 = Function(VO)
uu12.vector()[:] = B1 * uu1.vector()
g2 = Function(V2)
g2.vector()[:] = BO2 * uu12.vector()
du1 = Function(V1)
n1 = FacetNormal(self.mesh1)
du1 = assemble(inner(grad(uu1), n1) * ds)
du1 = project(du1, V1)
du12 = Function(VO)
du12.vector()[:] = B1 * du1.vector()
f2 = Function(V2)
f2.vector()[:] = BO2 * du12.vector()
#Step 4: solve for u2
boundary = right_robin
K2, _ = self.assemble_lui_stiffness(g=g2,
f=f2,
mesh=self.mesh2,
robin_boundary=boundary)
bc2 = DirichletBC(V2, Constant(0.), right_dirichlet)
bc2.apply(K2)
solver2 = standard_solver(K2, M2)
solver2.solve(10)
r2, _, rx2, _ = solver2.get_eigenpair(mode_number)
uu2 = Function(V2)
if (np.absolute(rx2.max()) < np.absolute(rx2.min())):
rx2 = self.adjustment * rx2 / (rx2.min())
uu2.vector()[:] = rx2
else:
rx2 = self.adjustment * rx2 / (rx2.max())
uu2.vector()[:] = rx2
#Step 5: Calculate the L2 norm for convergence
uu21.vector()[:] = B2 * uu2.vector()
uu12.vector()[:] = B1 * uu1.vector()
error = ((uu12 - uu21)**2) * dx
L2 = assemble(error)
semi_error = inner(
grad(uu12) - grad(uu21),
grad(uu12) - grad(uu21)) * dx
H1 = L2 + assemble(semi_error)
iter += 1
r_left[iter] = r1
r_right[iter] = r2
L2_error[iter] = L2
H1_error[iter] = H1
SH_error[iter] = H1 - L2
return L2_error, H1_error, SH_error, uu1, uu2, r_left, r_right
u0 = Constant("0.0")
bc = DirichletBC(V, u0, u0_boundary)
# Define target medium and rhs:
mtrue_exp = Expression('1 + 7*(pow(pow(x[0] - 0.5,2) +' + \
' pow(x[1] - 0.5,2),0.5) > 0.2)')
#mtrue = interpolate(mtrue_exp, Vme)
mtrue = interpolate(mtrue_exp, Vm)
f = Expression("1.0")
# Assemble weak form
trial = TrialFunction(V)
test = TestFunction(V)
a_true = inner(mtrue*nabla_grad(trial), nabla_grad(test))*dx
A_true = assemble(a_true)
bc.apply(A_true)
solver = PETScKrylovSolver('cg') # doesn't work with ilu preconditioner
#solver = LUSolver() # doesn't work in parallel !?
#solver.parameters['reuse_factorization'] = True
solver.set_operator(A_true)
# Assemble rhs
L = f*test*dx
b = assemble(L)
bc.apply(b)
# Solve:
u_true = Function(V)
"""
solver.solve(u_true.vector(), b)
if myrank == 0: print 'By hand:\n'
print 'P{0}: max(u)={1}\n'.format(myrank, max(u_true.vector().array()))
def test_krylov_reuse_pc():
"Test preconditioner re-use with PETScKrylovSolver"
# Define problem
mesh = UnitSquareMesh(MPI.comm_world, 8, 8)
V = FunctionSpace(mesh, ('Lagrange', 1))
bc = DirichletBC(V, Constant(0.0), lambda x, on_boundary: on_boundary)
u = TrialFunction(V)
v = TestFunction(V)
# Forms
a, L = inner(grad(u), grad(v)) * dx, dot(Constant(1.0), v) * dx
A, P = PETScMatrix(), PETScMatrix()
b = PETScVector()
# Assemble linear algebra objects
assemble(a, tensor=A) # noqa
assemble(a, tensor=P) # noqa
assemble(L, tensor=b) # noqa
# Apply boundary conditions
bc.apply(A)
bc.apply(P)
bc.apply(b)
# Create Krysolv solver and set operators
solver = PETScKrylovSolver("gmres", "bjacobi")
solver.set_operators(A, P)
# Solve
x = PETScVector()
num_iter_ref = solver.solve(x, b)
# Change preconditioner matrix (bad matrix) and solve (PC will be
# updated)
a_p = u * v * dx
assemble(a_p, tensor=P) # noqa
bc.apply(P)
x = PETScVector()
num_iter_mod = solver.solve(x, b)
assert num_iter_mod > num_iter_ref
# Change preconditioner matrix (good matrix) and solve (PC will be
# updated)
a_p = a
assemble(a_p, tensor=P) # noqa
bc.apply(P)
x = PETScVector()
num_iter = solver.solve(x, b)
assert num_iter == num_iter_ref
# Change preconditioner matrix (bad matrix) and solve (PC will not
# be updated)
solver.set_reuse_preconditioner(True)
a_p = u * v * dx
assemble(a_p, tensor=P) # noqa
bc.apply(P)
x = PETScVector()
num_iter = solver.solve(x, b)
assert num_iter == num_iter_ref
# Update preconditioner (bad PC, will increase iteration count)
solver.set_reuse_preconditioner(False)
x = PETScVector()
num_iter = solver.solve(x, b)
assert num_iter == num_iter_mod
def solverNeilanSalgadoZhang(P, opt):
'''
This function implements the method presented in
Neilan, Salgado, Zhang (2017), Chapter 4
Main characteristics:
- test with piecewise second derivative of test_u
- no discrete Hessian
- first-order stabilization necessary
'''
assert opt["HessianSpace"] == "CG", 'opt["HessianSpace"] has to be "CG"'
assert opt[
"stabilityConstant1"], 'opt["stabilityConstant1"] has to be positive'
gamma = P.normalizeSystem(opt)
if isinstance(P.g, list):
bc_V = []
for fun, dom in P.g:
bc_V.append(DirichletBC(P.V, fun, dom))
else:
if isinstance(P.g, Function):
bc_V = DirichletBC(P.V, P.g, 'on_boundary')
else:
g = project(P.g, P.V)
bc_V = DirichletBC(P.V, g, 'on_boundary')
trial_u = TrialFunction(P.V)
test_u = TestFunction(P.V)
# Assemble right-hand side in each case
f_h = gamma * P.f * div(grad(test_u)) * dx
# Adjust load vector in case of time-dependent problem
if P.isTimeDependant:
f_h = gamma * P.u_np1 * div(grad(test_u)) * dx - P.dt * f_h
rhs = assemble(f_h)
if isinstance(bc_V, list):
for bc_v in bc_V:
bc_v.apply(rhs)
else:
bc_V.apply(rhs)
repeat_setup = False
if hasattr(P, 'timeDependentCoefs'):
if P.timeDependentCoefs:
repeat_setup = True
if 'HJB' in str(type(P)):
repeat_setup = True
if (not P.isTimeDependant) or (P.iter == 0) or repeat_setup:
print('Setup bilinear form')
# Define bilinear form
a_h = gamma * inner(P.a, grad(grad(trial_u))) * div(grad(test_u)) * dx \
+ opt["stabilityConstant1"] * avg(P.hE)**(-1) * inner(
jump(grad(trial_u), P.nE), jump(grad(test_u), P.nE)) * dS
if P.hasDrift:
a_h += gamma * inner(P.b, grad(trial_u)) * div(grad(test_u)) * dx
if P.hasPotential:
a_h += gamma * P.c * trial_u * div(grad(test_u)) * dx
# Adjust system matrix in case of time-dependent problem
if P.isTimeDependant:
a_h = gamma * trial_u * div(grad(test_u)) * dx - P.dt * a_h
S = assemble(a_h)
if isinstance(bc_V, list):
for bc_v in bc_V:
bc_v.apply(S)
else:
bc_V.apply(S)
P.S = S
if opt['time_check']:
t1 = time()
solve(P.S, P.u.vector(), rhs)
# tmp = assemble(inner(P.a,grad(grad(trial_u))) * div(grad(test_u)) * dx)
# A = assemble(a_h)
# print('Row sum: ', sum(A.array(),1).round(4))
# print('Col sum: ', sum(A.array(),0).round(4))
# print('Total sum: ', sum(sum(A.array())).round(4))
# ipdb.set_trace()
# solve(a_h == f_h, P.u, bc_V, solver_parameters={'linear_solver': 'mumps'})
if opt['time_check']:
print("Solve linear equation system ... %.2fs" % (time() - t1))
sys.stdout.flush()
N_iter = 1
return N_iter
def test_DoGIP_vs_FEniCS(self):
print(
'\n== testing DoGIP vs. FEniCS for problem of weighted projection ===='
)
for dim, pol_order in itertools.product([2, 3], [1, 2]):
print('dim={}; pol_order={}'.format(dim, pol_order))
N = 2 # no. of elements
# creating MESH, defining MATERIAL and SOURCE
if dim == 2:
mesh = UnitSquareMesh(N, N)
m = Expression("1+10*16*x[0]*(1-x[0])*x[1]*(1-x[1])", degree=4)
f = Expression("80*x[0]*(0.5-x[0])*(1.-x[0])*x[1]*(1.-x[1])",
degree=5)
elif dim == 3:
mesh = UnitCubeMesh(N, N, N)
m = Expression("1+10*16*x[0]*(1-x[0])*(1-x[1])*x[2]", degree=4)
f = Expression("80*x[0]*(0.5-x[0])*(1.-x[0])*x[1]*(1.-x[1])",
degree=5)
mesh.coordinates()[:] += 0.1 * np.random.random(
mesh.coordinates().shape) # mesh perturbation
## standard approach with FEniCS #############################################
V = FunctionSpace(mesh, "CG", pol_order) # original FEM space
bc = DirichletBC(V, Constant(0.0),
lambda x, on_boundary: on_boundary)
u, v = TrialFunction(V), TestFunction(V)
u_fenics = Function(V) # the vector for storing the solution
solve(m * inner(grad(u), grad(v)) * dx == f * v * dx, u_fenics,
bc) # solution by FEniCS
## DoGIP - double-grid integration with interpolation-projection #############
W = FunctionSpace(mesh, "DG",
2 * (pol_order - 1)) # double-grid space
Wvector = VectorFunctionSpace(
mesh, "DG",
2 * (pol_order - 1)) # vector variant of double-grid space
w = TestFunction(W)
A_dogip = assemble(
m * w *
dx).get_local() # diagonal matrix of material coefficients
A_dogip_full = np.einsum(
'i,jk->ijk', A_dogip,
np.eye(dim)) # block-diagonal mat. for non-isotropic mat.
bv = assemble(f * v * dx)
bc.apply(bv)
b = bv.get_local() # vector of right-hand side
# assembling global interpolation-projection matrix B
B = get_B(V, Wvector, problem=1)
# solution to DoGIP problem
def Afun(x):
Axd = np.einsum('...jk,...j', A_dogip_full,
B.dot(x).reshape((-1, dim)))
Afunx = B.T.dot(Axd.ravel())
Afunx[list(bc.get_boundary_values()
)] = 0 # application of Dirichlet BC
return Afunx
Alinoper = linalg.LinearOperator((b.size, b.size),
matvec=Afun,
dtype=np.float) # system matrix
x, info = linalg.cg(Alinoper,
b,
x0=np.zeros_like(b),
tol=1e-8,
maxiter=1e2) # conjugate gradients
# testing the difference between DoGIP and FEniCS
self.assertAlmostEqual(
0, np.linalg.norm(u_fenics.vector().get_local() - x))
print('...ok')
def forward(mu_expression,
lmbda_expression,
rho,
Lx=10,
Ly=10,
t_end=1,
omega_p=5,
amplitude=5000,
center=0,
target=False):
Lpml = Lx / 10
#c_p = cp(mu.vector(), lmbda.vector(), rho)
max_velocity = 200 #c_p.max()
stable_hx = stable_dx(max_velocity, omega_p)
nx = int(Lx / stable_hx) + 1
#nx = max(nx, 60)
ny = int(Ly * nx / Lx) + 1
mesh = mesh_generator(Lx, Ly, Lpml, nx, ny)
used_hx = Lx / nx
dt = stable_dt(used_hx, max_velocity)
cfl_ct = cfl_constant(max_velocity, dt, used_hx)
print(used_hx, stable_hx)
print(cfl_ct)
#time.sleep(10)
PE = FunctionSpace(mesh, "DG", 0)
mu = interpolate(mu_expression, PE)
lmbda = interpolate(lmbda_expression, PE)
m = 2
R = 10e-8
t = 0.0
gamma = 0.50
beta = 0.25
ff = MeshFunction("size_t", mesh, mesh.geometry().dim() - 1)
Dirichlet(Lx, Ly, Lpml).mark(ff, 1)
# Create function spaces
VE = VectorElement("CG", mesh.ufl_cell(), 1, dim=2)
TE = TensorElement("DG", mesh.ufl_cell(), 0, shape=(2, 2), symmetry=True)
W = FunctionSpace(mesh, MixedElement([VE, TE]))
F = FunctionSpace(mesh, "CG", 2)
V = W.sub(0).collapse()
M = W.sub(1).collapse()
alpha_0 = Alpha_0(m, stable_hx, R, Lpml)
alpha_1 = Alpha_1(alpha_0, Lx, Lpml, degree=2)
alpha_2 = Alpha_2(alpha_0, Ly, Lpml, degree=2)
beta_0 = Beta_0(m, max_velocity, R, Lpml)
beta_1 = Beta_1(beta_0, Lx, Lpml, degree=2)
beta_2 = Beta_2(beta_0, Ly, Lpml, degree=2)
alpha_1 = interpolate(alpha_1, F)
alpha_2 = interpolate(alpha_2, F)
beta_1 = interpolate(beta_1, F)
beta_2 = interpolate(beta_2, F)
a_ = alpha_1 * alpha_2
b_ = alpha_1 * beta_2 + alpha_2 * beta_1
c_ = beta_1 * beta_2
Lambda_e = as_tensor([[alpha_2, 0], [0, alpha_1]])
Lambda_p = as_tensor([[beta_2, 0], [0, beta_1]])
a_ = alpha_1 * alpha_2
b_ = alpha_1 * beta_2 + alpha_2 * beta_1
c_ = beta_1 * beta_2
Lambda_e = as_tensor([[alpha_2, 0], [0, alpha_1]])
Lambda_p = as_tensor([[beta_2, 0], [0, beta_1]])
# Set up boundary condition
bc = DirichletBC(W.sub(0), Constant(("0.0", "0.0")), ff, 1)
# Create measure for the source term
dx = Measure("dx", domain=mesh)
ds = Measure("ds", domain=mesh, subdomain_data=ff)
# Set up initial values
u0 = Function(V)
u0.set_allow_extrapolation(True)
v0 = Function(V)
a0 = Function(V)
U0 = Function(M)
V0 = Function(M)
A0 = Function(M)
# Test and trial functions
(u, S) = TrialFunctions(W)
(w, T) = TestFunctions(W)
g = ModifiedRickerPulse(0, omega_p, amplitude, center)
F = rho * inner(a_ * N_ddot(u, u0, a0, v0, dt, beta) \
+ b_ * N_dot(u, u0, v0, a0, dt, beta, gamma) + c_ * u, w) * dx \
+ inner(N_dot(S, U0, V0, A0, dt, beta, gamma).T * Lambda_e + S.T * Lambda_p, grad(w)) * dx \
- inner(g, w) * ds \
+ inner(compliance(a_ * N_ddot(S, U0, A0, V0, dt, beta) + b_ * N_dot(S, U0, V0, A0, dt, beta, gamma) + c_ * S, u, mu, lmbda), T) * dx \
- 0.5 * inner(grad(u) * Lambda_p + Lambda_p * grad(u).T + grad(N_dot(u, u0, v0, a0, dt, beta, gamma)) * Lambda_e \
+ Lambda_e * grad(N_dot(u, u0, v0, a0, dt, beta, gamma)).T, T) * dx \
a, L = lhs(F), rhs(F)
# Assemble rhs (once)
A = assemble(a)
# Create GMRES Krylov solver
solver = KrylovSolver(A, "gmres")
# Create solution function
S = Function(W)
if target:
xdmffile_u = XDMFFile("inversion_temporal_file/target/u.xdmf")
pvd = File("inversion_temporal_file/target/u.pvd")
xdmffile_u.write(u0, t)
timeseries_u = TimeSeries(
"inversion_temporal_file/target/u_timeseries")
else:
xdmffile_u = XDMFFile("inversion_temporal_file/obs/u.xdmf")
xdmffile_u.write(u0, t)
timeseries_u = TimeSeries("inversion_temporal_file/obs/u_timeseries")
rec_counter = 0
while t < t_end - 0.5 * dt:
t += float(dt)
if rec_counter % 10 == 0:
print(
'\n\rtime: {:.3f} (Progress: {:.2f}%)'.format(
t, 100 * t / t_end), )
g.t = t
# Assemble rhs and apply boundary condition
b = assemble(L)
bc.apply(A, b)
# Compute solution
solver.solve(S.vector(), b)
(u, U) = S.split(True)
# Update previous time step
update(u, u0, v0, a0, beta, gamma, dt)
update(U, U0, V0, A0, beta, gamma, dt)
xdmffile_u.write(u, t)
pvd << (u, t)
timeseries_u.store(u.vector(), t)
energy = inner(u, u) * dx
E = assemble(energy)
print("E = ", E)
print(u.vector().max())
for i, t in enumerate(trange[1:]):
print('Solve problem, time step t = {}'.format(np.round(t, 4)))
# Assign current time to variable t_
t_.assign(t)
# Compute Ytilde, i.e. y-position of process starting in x_i,y_j
# after application of convection in y-direction
Ytilde = Y + by(t, X, Y) * dt[i]
Vtilde = Yinterp1(vold, Ytilde, ydofs, ydiff)
# Set up problem
Sfull = M - dt[i] * S
# Apply boundary conditions
bc_Vx.apply(Sfull)
# Loop over all layers in y-direction
for j in range(ny + 1):
y_.assign(ydofs[j])
L = assemble(L_)
# This is ordinary matrix multiplication
# np.max(np.abs(M*Vtilde[3,:] - M.array() @ Vtilde[3,:]))
Mvtmp.vector().set_local(M * Vtilde[j, :])
Lfull = Mvtmp.vector() - dt[i] * L
bc_Vx.apply(Lfull)
def solve(self):
""" Find eigenvalues for transformed mesh. """
self.progress("Building mesh.")
# build transformed mesh
mesh = self.refineMesh()
# dim = mesh.topology().dim()
if self.bcLast:
mesh = transform_mesh(mesh, self.transformList)
Robin, Steklov, shift, bcs = get_bc_parts(mesh, self.bcList)
else:
Robin, Steklov, shift, bcs = get_bc_parts(mesh, self.bcList)
mesh = transform_mesh(mesh, self.transformList)
# boundary conditions computed on non-transformed mesh
# copy the values to transformed mesh
fun = FacetFunction("size_t", mesh, shift)
fun.array()[:] = bcs.array()[:]
bcs = fun
ds = Measure('ds', domain=mesh, subdomain_data=bcs)
V = FunctionSpace(mesh, self.method, self.deg)
u = TrialFunction(V)
v = TestFunction(V)
self.progress("Assembling matrices.")
wTop = Expression(self.wTop)
wBottom = Expression(self.wBottom)
#
# build stiffness matrix form
#
s = dot(grad(u), grad(v)) * wTop * dx
# add Robin parts
for bc in Robin:
s += Constant(bc.parValue) * u * v * wTop * ds(bc.value + shift)
#
# build mass matrix form
#
if len(Steklov) > 0:
m = 0
for bc in Steklov:
m += Constant(
bc.parValue) * u * v * wBottom * ds(bc.value + shift)
else:
m = u * v * wBottom * dx
# assemble
# if USE_EIGEN:
# S, M = EigenMatrix(), EigenMatrix()
# tempv = EigenVector()
# else:
S, M = PETScMatrix(), PETScMatrix()
# tempv = PETScVector()
if not np.any(bcs.array() == shift + 1):
# no Dirichlet parts
assemble(s, tensor=S)
assemble(m, tensor=M)
else:
#
# with EIGEN we could
# apply Dirichlet condition symmetrically
# completely remove rows and columns
#
# Dirichlet parts are marked with shift+1
#
# temp = Constant(0)*v*dx
bc = DirichletBC(V, Constant(0.0), bcs, shift + 1)
# assemble_system(s, temp, bc, A_tensor=S, b_tensor=tempv)
# assemble_system(m, temp, bc, A_tensor=M, b_tensor=tempv)
assemble(s, tensor=S)
bc.apply(S)
assemble(m, tensor=M)
# bc.zero(M)
# if USE_EIGEN:
# M = M.sparray()
# M.eliminate_zeros()
# print M.shape
# indices = M.indptr[:-1] - M.indptr[1:] < 0
# M = M[indices, :].tocsc()[:, indices]
# S = S.sparray()[indices, :].tocsc()[:, indices]
# print M.shape
#
# solve the eigenvalue problem
#
self.progress("Solving eigenvalue problem.")
eigensolver = SLEPcEigenSolver(S, M)
eigensolver.parameters["problem_type"] = "gen_hermitian"
eigensolver.parameters["solver"] = "krylov-schur"
if self.target is not None:
eigensolver.parameters["spectrum"] = "target real"
eigensolver.parameters["spectral_shift"] = self.target
else:
eigensolver.parameters["spectrum"] = "smallest magnitude"
eigensolver.parameters["spectral_shift"] = -0.01
eigensolver.parameters["spectral_transform"] = "shift-and-invert"
eigensolver.solve(self.number)
self.progress("Generating eigenfunctions.")
if eigensolver.get_number_converged() == 0:
return None
eigf = []
eigv = []
if self.deg > 1:
mesh = refine(mesh)
W = FunctionSpace(mesh, 'CG', 1)
for i in range(eigensolver.get_number_converged()):
pair = eigensolver.get_eigenpair(i)[::2]
eigv.append(pair[0])
u = Function(V)
u.vector()[:] = pair[1]
eigf.append(interpolate(u, W))
return eigv, eigf
while t < t_end - 0.5 * dt:
t += float(dt)
if rank == 0 and rec_counter % 10 == 0:
print(
'\n\rtime: {:.3f} (Progress: {:.2f}%)'.format(
t, 100 * t / t_end), )
print("time taken til this point: {}".format(time.time() - t0))
for pulse in pulses:
pulse.t = t
g = sum(pulses)
# Assemble rhs and apply boundary condition
b = assemble(L)
bc.apply(A, b)
# Compute solution
solver.solve(S.vector(), b)
(u, U) = S.split(True)
# Update previous time step
update(u, u0, v0, a0, beta, gamma, dt)
update(U, U0, V0, A0, beta, gamma, dt)
pvd << (u0, t)
rec_counter += 1
energy = inner(u, u) * dx
E = assemble(energy)
print("E = ", E)
def __call__(self, degree=3, y=0., standard_deviation=1.):
V = FunctionSpace(self._mesh, "CG", degree)
v = TestFunction(V)
potential_trial = TrialFunction(V)
potential = Function(V)
dx = Measure("dx")(subdomain_data=self._subdomains)
# ds = Measure("ds")(subdomain_data=self._boundaries)
csd = Expression(f'''
x[1] >= {self.H} ?
0 :
a * exp({-0.5 / standard_deviation ** 2}
* ((x[0])*(x[0])
+ (x[1] - {y})*(x[1] - {y})
+ (x[2])*(x[2])
))
''',
degree=degree,
a=1.0)
self.a = csd.a = 1.0 / assemble(
csd * Measure("dx", self._mesh))
# print(assemble(csd * Measure("dx", self._mesh)))
L = csd * v * dx
known_terms = assemble(L)
# a = (inner(grad(potential_trial), grad(v))
# * (Constant(self.SALINE_CONDUCTIVITY) * dx(self.SALINE_VOL)
# + Constant(self.SLICE_CONDUCTIVITY) * (dx(self.SLICE_VOL)
# + dx(self.ROI_VOL))))
a = sum(
Constant(conductivity) *
inner(grad(potential_trial), grad(v)) * dx(domain)
for domain, conductivity in [
(self.SALINE_VOL, self.SALINE_CONDUCTIVITY),
(self.SLICE_VOL, self.SLICE_CONDUCTIVITY),
(self.ROI_VOL, self.SLICE_CONDUCTIVITY),
])
terms_with_unknown = assemble(a)
dirchlet_bc = DirichletBC(
V,
Constant(2.0 * 0.25 /
(self.RADIUS * np.pi * self.SALINE_CONDUCTIVITY)),
# 2.0 becaue of dielectric base duplicating
# the current source
# slice conductivity and thickness considered
# negligible
self._boundaries,
self.MODEL_DOME)
dirchlet_bc.apply(terms_with_unknown, known_terms)
solver = KrylovSolver("cg", "ilu")
solver.parameters["maximum_iterations"] = MAX_ITER
solver.parameters["absolute_tolerance"] = 1E-8
# solver.parameters["monitor_convergence"] = True
start = datetime.datetime.now()
try:
self.iterations = solver.solve(terms_with_unknown,
potential.vector(),
known_terms)
return potential
except RuntimeError as e:
self.iterations = MAX_ITER
logger.warning("Solver failed: {}".format(repr(e)))
return None
finally:
self.time = datetime.datetime.now() - start