def amg_solve(N, method):
# Elasticity parameters
E = 1.0e9
nu = 0.3
mu = E / (2.0 * (1.0 + nu))
lmbda = E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu))
# Stress computation
def sigma(v):
return 2.0 * mu * sym(grad(v)) + lmbda * tr(sym(
grad(v))) * Identity(2)
# Define problem
mesh = UnitSquareMesh(MPI.comm_world, N, N)
V = VectorFunctionSpace(mesh, 'Lagrange', 1)
bc0 = Function(V)
with bc0.vector().localForm() as bc_local:
bc_local.set(0.0)
def boundary(x, only_boundary):
return [only_boundary] * x.shape(0)
bc = DirichletBC(V.sub(0), bc0, boundary)
u = TrialFunction(V)
v = TestFunction(V)
# Forms
a, L = inner(sigma(u), grad(v)) * dx, dot(ufl.as_vector(
(1.0, 1.0)), v) * dx
# Assemble linear algebra objects
A = assemble_matrix(a, [bc])
A.assemble()
b = assemble_vector(L)
apply_lifting(b, [a], [[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
# Create solution function
u = Function(V)
# Create near null space basis and orthonormalize
null_space = build_nullspace(V, u.vector())
# Attached near-null space to matrix
A.set_near_nullspace(null_space)
# Test that basis is orthonormal
assert null_space.is_orthonormal()
# Create PETSC smoothed aggregation AMG preconditioner, and
# create CG solver
solver = PETScKrylovSolver("cg", method)
# Set matrix operator
solver.set_operator(A)
# Compute solution and return number of iterations
return solver.solve(u.vector(), b)
def fit(x0, y0, mesh, Eps, degree=1, verbose=False, solver='spsolve'):
V = FunctionSpace(mesh, 'CG', degree)
u = TrialFunction(V)
v = TestFunction(V)
n = FacetNormal(mesh)
dim = mesh.geometry().dim()
A = [
_assemble_eigen(+Constant(Eps[i, j]) * u.dx(i) * v.dx(j) * dx
# pylint: disable=unsubscriptable-object
- Constant(Eps[i, j]) * u.dx(i) * n[j] * v *
ds).sparray() for i in range(dim) for j in range(dim)
]
E = _build_eval_matrix(V, x0)
M = sparse.vstack(A + [E])
b = numpy.concatenate([numpy.zeros(sum(a.shape[0] for a in A)), y0])
if solver == 'spsolve':
MTM = M.T.dot(M)
x = sparse.linalg.spsolve(MTM, M.T.dot(b))
elif solver == 'lsqr':
x, istop, *_ = sparse.linalg.lsqr(
M,
b,
show=verbose,
atol=1.0e-10,
btol=1.0e-10,
)
assert istop == 2, \
'sparse.linalg.lsqr not successful (error code {})'.format(istop)
elif solver == 'lsmr':
x, istop, *_ = sparse.linalg.lsmr(
M,
b,
show=verbose,
atol=1.0e-10,
btol=1.0e-10,
# min(M.shape) is the default
maxiter=max(min(M.shape), 10000))
assert istop == 2, \
'sparse.linalg.lsmr not successful (error code {})'.format(istop)
else:
assert solver == 'gmres', 'Unknown solver \'{}\'.'.format(solver)
A = sparse.linalg.LinearOperator((M.shape[1], M.shape[1]),
matvec=lambda x: M.T.dot(M.dot(x)))
x, info = sparse.linalg.gmres(A, M.T.dot(b), tol=1.0e-12)
assert info == 0, \
'sparse.linalg.gmres not successful (error code {})'.format(info)
u = Function(V)
u.vector().set_local(x)
return u
def amg_solve(N, method):
# Elasticity parameters
E = 1.0e9
nu = 0.3
mu = E / (2.0 * (1.0 + nu))
lmbda = E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu))
# Stress computation
def sigma(v):
return 2.0 * mu * sym(grad(v)) + lmbda * tr(sym(
grad(v))) * Identity(2)
# Define problem
mesh = UnitSquareMesh(MPI.comm_world, N, N)
V = VectorFunctionSpace(mesh, 'Lagrange', 1)
bc0 = Function(V)
bc = DirichletBC(V.sub(0), bc0, lambda x, on_boundary: on_boundary)
u = TrialFunction(V)
v = TestFunction(V)
# Forms
a, L = inner(sigma(u), grad(v)) * dx, dot(ufl.as_vector(
(1.0, 1.0)), v) * dx
# Assemble linear algebra objects
A, b = assemble_system(a, L, bc)
# Create solution function
u = Function(V)
# Create near null space basis and orthonormalize
null_space = build_nullspace(V, u.vector())
# Attached near-null space to matrix
A.set_near_nullspace(null_space)
# Test that basis is orthonormal
assert null_space.is_orthonormal()
# Create PETSC smoothed aggregation AMG preconditioner, and
# create CG solver
solver = PETScKrylovSolver("cg", method)
# Set matrix operator
solver.set_operator(A)
# Compute solution and return number of iterations
return solver.solve(u.vector(), b)
def solve(self, mesh, num=5):
""" Solve for num eigenvalues based on the mesh. """
# conforming elements
V = FunctionSpace(mesh, "CG", self.degree)
u = TrialFunction(V)
v = TestFunction(V)
# weak formulation
a = inner(grad(u), grad(v)) * dx
b = u * v * ds
A = PETScMatrix()
B = PETScMatrix()
A = assemble(a, tensor=A)
B = assemble(b, tensor=B)
# find eigenvalues
eigensolver = SLEPcEigenSolver(A, B)
eigensolver.parameters["spectral_transform"] = "shift-and-invert"
eigensolver.parameters["problem_type"] = "gen_hermitian"
eigensolver.parameters["spectrum"] = "smallest real"
eigensolver.parameters["spectral_shift"] = 1.0E-10
eigensolver.solve(num + 1)
# extract solutions
lst = [
eigensolver.get_eigenpair(i)
for i in range(1, eigensolver.get_number_converged())
]
for k in range(len(lst)):
u = Function(V)
u.vector()[:] = lst[k][2]
lst[k] = (lst[k][0], u) # pair (eigenvalue,eigenfunction)
return np.array(lst)
def solve(self, mesh, num=5):
""" Solve for num eigenvalues based on the mesh. """
# conforming elements
V = FunctionSpace(mesh, "CG", self.degree)
u = TrialFunction(V)
v = TestFunction(V)
# weak formulation
a = inner(grad(u), grad(v)) * dx
b = u * v * ds
A = PETScMatrix()
B = PETScMatrix()
A = assemble(a, tensor=A)
B = assemble(b, tensor=B)
# find eigenvalues
eigensolver = SLEPcEigenSolver(A, B)
eigensolver.parameters["spectral_transform"] = "shift-and-invert"
eigensolver.parameters["problem_type"] = "gen_hermitian"
eigensolver.parameters["spectrum"] = "smallest real"
eigensolver.parameters["spectral_shift"] = 1.0E-10
eigensolver.solve(num + 1)
# extract solutions
lst = [
eigensolver.get_eigenpair(i) for i in range(
1,
eigensolver.get_number_converged())]
for k in range(len(lst)):
u = Function(V)
u.vector()[:] = lst[k][2]
lst[k] = (lst[k][0], u) # pair (eigenvalue,eigenfunction)
return np.array(lst)
def cm1dvelocity(img: np.array, vel: np.array, alpha0: float,
alpha1: float) -> np.array:
"""Computes the source for a L2-H1 mass conserving flow for a 1D image
sequence and a given velocity.
Takes a one-dimensional image sequence and a velocity, and returns a
minimiser of the L2-H1 mass conservation functional with spatio-temporal
regularisation.
Args:
img (np.array): A 1D image sequence of shape (m, n), where m is the
number of time steps and n is the number of pixels.
vel (np.array): A 1D image sequence of shape (m, n).
alpha0 (float): The spatial regularisation parameter.
alpha1 (float): The temporal regularisation parameter.
Returns:
k: A source array of shape (m, n).
"""
# Create mesh.
[m, n] = img.shape
mesh = UnitSquareMesh(m - 1, n - 1)
# Define function space and functions.
V = FunctionSpace(mesh, 'CG', 1)
k = TrialFunction(V)
w = TestFunction(V)
# Convert image to function.
f = Function(V)
f.vector()[:] = dh.img2funvec(img)
# Convert velocity to function.
v = Function(V)
v.vector()[:] = dh.img2funvec(vel)
# Define derivatives of data.
ft = Function(V)
ftv = np.diff(img, axis=0) * (m - 1)
ftv = np.concatenate((ftv, ftv[-1, :].reshape(1, n)), axis=0)
ft.vector()[:] = dh.img2funvec(ftv)
fx = Function(V)
fxv = np.gradient(img, 1 / (n - 1), axis=1)
fx.vector()[:] = dh.img2funvec(fxv)
# Define weak formulation.
A = k * w * dx + alpha0 * k.dx(1) * w.dx(1) * dx + alpha1 * k.dx(0) * w.dx(
0) * dx
b = (ft + v.dx(1) * f + v * fx) * w * dx
# Compute solution.
k = Function(V)
solve(A == b, k)
# Convert back to array.
k = dh.funvec2img(k.vector().get_local(), m, n)
return k
def cm1dsource(img: np.array, k: np.array, alpha0: float,
alpha1: float) -> np.array:
"""Computes the L2-H1 mass conserving flow for a 1D image sequence and a
given source.
Takes a one-dimensional image sequence and a source, and returns a
minimiser of the L2-H1 mass conservation functional with spatio-temporal
regularisation.
Args:
img (np.array): A 1D image sequence of shape (m, n), where m is the
number of time steps and n is the number of pixels.
k (np.array): A 1D image sequence of shape (m, n).
alpha0 (float): The spatial regularisation parameter.
alpha1 (float): The temporal regularisation parameter.
Returns:
v: A velocity array of shape (m, n).
"""
# Create mesh.
[m, n] = img.shape
mesh = UnitSquareMesh(m - 1, n - 1)
# Define function space and functions.
V = FunctionSpace(mesh, 'CG', 1)
v = TrialFunction(V)
w = TestFunction(V)
# Convert image to function.
f = Function(V)
f.vector()[:] = dh.img2funvec(img)
# Convert source to function.
g = Function(V)
g.vector()[:] = dh.img2funvec(k)
# Define derivatives of data.
ft = Function(V)
ftv = np.diff(img, axis=0) * (m - 1)
ftv = np.concatenate((ftv, ftv[-1, :].reshape(1, n)), axis=0)
ft.vector()[:] = dh.img2funvec(ftv)
fx = Function(V)
fxv = np.gradient(img, 1 / (n - 1), axis=1)
fx.vector()[:] = dh.img2funvec(fxv)
ft = f.dx(0)
fx = f.dx(1)
# Define weak formulation.
A = - (fx*v + f*v.dx(1)) * (fx*w + f*w.dx(1))*dx \
- alpha0*v.dx(1)*w.dx(1)*dx - alpha1*v.dx(0)*w.dx(0)*dx
b = ft * (fx * w + f * w.dx(1)) * dx - g * (fx * w + f * w.dx(1)) * dx
# Compute solution.
v = Function(V)
solve(A == b, v)
# Convert back to array.
vel = dh.funvec2img(v.vector().get_local(), m, n)
return vel
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 _fit_dolfin(x0,
y0,
points,
cells,
lmbda: float,
degree: int = 1,
solver: str = "lsqr"):
from dolfin import (
BoundingBoxTree,
Cell,
EigenMatrix,
FacetNormal,
Function,
FunctionSpace,
Mesh,
MeshEditor,
Point,
TestFunction,
TrialFunction,
assemble,
dot,
ds,
dx,
grad,
)
def _assemble_eigen(form):
L = EigenMatrix()
assemble(form, tensor=L)
return L
def _build_eval_matrix(V, points):
"""Build the sparse m-by-n matrix that maps a coefficient set for a function in
V to the values of that function at m given points."""
# See <https://www.allanswered.com/post/lkbkm/#zxqgk>
mesh = V.mesh()
bbt = BoundingBoxTree()
bbt.build(mesh)
dofmap = V.dofmap()
el = V.element()
sdim = el.space_dimension()
rows = []
cols = []
data = []
for i, x in enumerate(points):
cell_id = bbt.compute_first_entity_collision(Point(*x))
cell = Cell(mesh, cell_id)
coordinate_dofs = cell.get_vertex_coordinates()
rows.append(np.full(sdim, i))
cols.append(dofmap.cell_dofs(cell_id))
v = el.evaluate_basis_all(x, coordinate_dofs, cell_id)
data.append(v)
rows = np.concatenate(rows)
cols = np.concatenate(cols)
data = np.concatenate(data)
m = len(points)
n = V.dim()
matrix = sparse.csr_matrix((data, (rows, cols)), shape=(m, n))
return matrix
editor = MeshEditor()
mesh = Mesh()
# Convert points, cells to dolfin mesh
if cells.shape[1] == 2:
editor.open(mesh, "interval", 1, 1, 1)
else:
# can only handle triangles for now
assert cells.shape[1] == 3
# topological and geometrical dimension 2
editor.open(mesh, "triangle", 2, 2, 1)
editor.init_vertices(len(points))
editor.init_cells(len(cells))
for k, point in enumerate(points):
editor.add_vertex(k, point)
for k, cell in enumerate(cells.astype(np.uintp)):
editor.add_cell(k, cell)
editor.close()
V = FunctionSpace(mesh, "CG", degree)
u = TrialFunction(V)
v = TestFunction(V)
mesh = V.mesh()
n = FacetNormal(mesh)
# omega = assemble(1 * dx(mesh))
A = _assemble_eigen(dot(grad(u), grad(v)) * dx -
dot(n, grad(u)) * v * ds).sparray()
A *= lmbda
E = _build_eval_matrix(V, x0)
# mass matrix
M = _assemble_eigen(u * v * dx).sparray()
x = _solve(A, M, E, y0, solver)
u = Function(V)
u.vector().set_local(x)
return u
def fit(x0, y0, V, lmbda, solver, prec_dirichlet_indices=None):
"""We're trying to minimize
sum_i w_i (f(xi) - yi)^2 + ||lmbda Delta f||^2_{L^2(Omega)}
over all functions f from V with weights w_i, lmbda. The discretization of this is
||W(E(f) - y)||_2^2 + ||lmbda Delta_h f_h||^2_{M^{-1}}
where E is the (small and fat) evaluation operator at coordinates x_i, Delta_h is
the discretization of Delta, and M is the mass matrix. One can either try and
minimize this equation with a generic method or solve the linear equation
lmbda A.T M^{-1} A x + E.T E x = E.T y0
for the extremum x. Unfortunately, solving the linear equation is not
straightforward. M is spd, A is nonsymmetric and rank-deficient but
positive-semidefinite. So far, we simply use sparse CG, but a good idea for a
preconditioner is highly welcome.
"""
u = TrialFunction(V)
v = TestFunction(V)
mesh = V.mesh()
n = FacetNormal(mesh)
# omega = assemble(1 * dx(mesh))
A = _assemble_eigen(dot(grad(u), grad(v)) * dx -
dot(n, grad(u)) * v * ds).sparray()
A *= lmbda
E = _build_eval_matrix(V, x0)
# mass matrix
M = _assemble_eigen(u * v * dx).sparray()
# Scipy implementations of both LSQR and LSMR can only be used with the standard l_2
# inner product. This is not sufficient here: We need the M inner product to make
# sure that the discrete residual is an approximation to the inner product of the
# continuous problem.
if solver == "dense-direct":
# Minv is dense, yikes!
a = A.toarray()
m = M.toarray()
e = E.toarray()
AT_Minv_A = numpy.dot(a.T, numpy.linalg.solve(m, a)) + numpy.dot(
e.T, e)
ET_b = numpy.dot(e.T, y0)
x = numpy.linalg.solve(AT_Minv_A, ET_b)
elif solver == "sparse-cg":
def matvec(x):
Ax = A.dot(x)
return A.T.dot(sparse.linalg.spsolve(M, Ax)) + E.T.dot(E.dot(x))
lop = pykry.LinearOperator((E.shape[1], E.shape[1]), float, dot=matvec)
ET_b = E.T.dot(y0)
out = pykry.cg(lop, ET_b, tol=1.0e-10, maxiter=1000)
x = out.xk
# import matplotlib.pyplot as plt
# plt.semilogy(out.resnorms)
# plt.grid()
# plt.show()
else:
def f(x):
Ax = A.dot(x)
Exy = E.dot(x) - y0
return numpy.dot(Ax, spsolve(M, Ax)) + numpy.dot(Exy, Exy)
# Set x0 to be the average of y0
x0 = numpy.full(A.shape[0], numpy.sum(y0) / y0.shape[0])
out = scipy.optimize.minimize(f, x0, method=solver)
x = out.x
u = Function(V)
u.vector().set_local(x)
return u
def femsolve():
''' Bilineaarinen muoto:
a(u,v) = L(v)
a(u,v) = (inner(grad(u), grad(v)) + u*v)*dx
L(v) = f*v*dx - g*v*ds
g(x) = -du/dx = -u1, x = x1
u(x0) = u0
Omega = {xeR|x0<=x<=x1}
'''
from dolfin import UnitInterval, FunctionSpace, DirichletBC, TrialFunction
from dolfin import TestFunction, grad, Constant, Function, solve, inner, dx, ds
from dolfin import MeshFunction, assemble
import dolfin
# from dolfin import set_log_level, PROCESS
# Create mesh and define function space
mesh = UnitInterval(30)
V = FunctionSpace(mesh, 'Lagrange', 2)
boundaries = MeshFunction('uint', mesh, mesh.topology().dim()-1)
boundaries.set_all(0)
class Left(dolfin.SubDomain):
def inside(self, x, on_boundary):
tol = 1E-14 # tolerance for coordinate comparisons
return on_boundary and abs(x[0]) < tol
class Right(dolfin.SubDomain):
def inside(self, x, on_boundary):
return dolfin.near(x[0], 1.0)
left = Left()
right = Right()
left.mark(boundaries, 1)
right.mark(boundaries, 2)
# def u0_boundary(x):
# return abs(x[0]) < tol
#
# bc = DirichletBC(V, Constant(u0), lambda x: abs(x[0]) < tol)
bcs = [DirichletBC(V, Constant(u0), boundaries, 1)]
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = (inner(grad(u), grad(v)) + u*v)*dx
g = Constant(-u1)
L = Constant(f)*v*dx - g*v*ds(2)
# set_log_level(PROCESS)
# Compute solution
A = assemble(a, exterior_facet_domains=boundaries)
b = assemble(L, exterior_facet_domains=boundaries)
for bc in bcs:
bc.apply(A, b)
u = Function(V)
solve(A, u.vector(), b, 'lu')
coor = mesh.coordinates()
u_array = u.vector().array()
a = []
b = []
for i in range(mesh.num_vertices()):
a.append(coor[i])
b.append(u_array[i])
print('u(%3.2f) = %0.14E'%(coor[i],u_array[i]))
import numpy as np
np.savez('fem',a,b)
# Improve estimate of eigenvalues for Chebyshev smoothing
PETScOptions.set("mg_levels_esteig_ksp_type", "cg")
PETScOptions.set("mg_levels_ksp_chebyshev_esteig_steps", 20)
# Monitor solver
PETScOptions.set("ksp_monitor")
# Create CG Krylov solver and turn convergence monitoring on
solver = PETScKrylovSolver(MPI.comm_world)
solver.set_from_options()
# Set matrix operator
solver.set_operator(A)
# Compute solution
solver.solve(u.vector(), b)
# Save solution to XDMF format
file = XDMFFile(MPI.comm_world, "elasticity.xdmf")
file.write(u)
unorm = u.vector().norm()
if MPI.rank(mesh.mpi_comm()) == 0:
print("Solution vector norm:", unorm)
# Save colored mesh partitions in VTK format if running in parallel
# if MPI.size(mesh.mpi_comm()) > 1:
# File("partitions.pvd") << MeshFunction("size_t", mesh, mesh.topology.dim, \
# MPI.rank(mesh.mpi_comm()))
# Project and write stress field to post-processing file
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
def femsolve():
''' Bilineaarinen muoto:
a(u,v) = L(v)
a(u,v) = (inner(grad(u), grad(v)) + u*v)*dx
L(v) = f*v*dx - g*v*ds
g(x) = -du/dx = -u1, x = x1
u(x0) = u0
Omega = {xeR|x0<=x<=x1}
'''
from dolfin import UnitInterval, FunctionSpace, DirichletBC, TrialFunction
from dolfin import TestFunction, grad, Constant, Function, solve, inner, dx, ds
from dolfin import MeshFunction, assemble
import dolfin
# from dolfin import set_log_level, PROCESS
# Create mesh and define function space
mesh = UnitInterval(30)
V = FunctionSpace(mesh, 'Lagrange', 2)
boundaries = MeshFunction('uint', mesh, mesh.topology().dim() - 1)
boundaries.set_all(0)
class Left(dolfin.SubDomain):
def inside(self, x, on_boundary):
tol = 1E-14 # tolerance for coordinate comparisons
return on_boundary and abs(x[0]) < tol
class Right(dolfin.SubDomain):
def inside(self, x, on_boundary):
return dolfin.near(x[0], 1.0)
left = Left()
right = Right()
left.mark(boundaries, 1)
right.mark(boundaries, 2)
# def u0_boundary(x):
# return abs(x[0]) < tol
#
# bc = DirichletBC(V, Constant(u0), lambda x: abs(x[0]) < tol)
bcs = [DirichletBC(V, Constant(u0), boundaries, 1)]
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = (inner(grad(u), grad(v)) + u * v) * dx
g = Constant(-u1)
L = Constant(f) * v * dx - g * v * ds(2)
# set_log_level(PROCESS)
# Compute solution
A = assemble(a, exterior_facet_domains=boundaries)
b = assemble(L, exterior_facet_domains=boundaries)
for bc in bcs:
bc.apply(A, b)
u = Function(V)
solve(A, u.vector(), b, 'lu')
coor = mesh.coordinates()
u_array = u.vector().array()
a = []
b = []
for i in range(mesh.num_vertices()):
a.append(coor[i])
b.append(u_array[i])
print('u(%3.2f) = %0.14E' % (coor[i], u_array[i]))
import numpy as np
np.savez('fem', a, b)
def run_simulation(
filepath,
topology_info: int = None,
top_bc: int = None,
bot_bc: int = None,
left_bc: int = None,
right_bc: int = None,
geometry: dict = None,
kappa=3, #only if geometry is None
show=True,
save_solution=False):
from dolfin import (Mesh, XDMFFile, MeshValueCollection, cpp,
FunctionSpace, TrialFunction, TestFunction,
DirichletBC, Constant, Measure, inner, nabla_grad,
Function, solve, plot, File)
mesh = Mesh()
with XDMFFile("%s_triangle.xdmf" % filepath.split('.')[0]) as infile:
infile.read(mesh) # read the complete mesh
mvc_subdo = MeshValueCollection("size_t", mesh,
mesh.geometric_dimension() - 1)
with XDMFFile("%s_triangle.xdmf" % filepath.split('.')[0]) as infile:
infile.read(mvc_subdo, "subdomains") # read the diferent subdomians
subdomains = cpp.mesh.MeshFunctionSizet(mesh, mvc_subdo)
mvc = MeshValueCollection("size_t", mesh, mesh.geometric_dimension() - 2)
with XDMFFile("%s_line.xdmf" % filepath.split('.')[0]) as infile:
infile.read(mvc, "boundary_conditions") #read the boundary conditions
boundary = cpp.mesh.MeshFunctionSizet(mesh, mvc)
# Define function space and basis functions
V = FunctionSpace(mesh, "CG", 1)
u = TrialFunction(V)
v = TestFunction(V)
# Boundary conditions
bcs = []
for bc_id in topology_info.keys():
if bc_id[-2:] == "bc":
if bot_bc is not None and bc_id[:3] == "bot":
bcs.append(
DirichletBC(V, Constant(bot_bc), boundary,
topology_info[bc_id]))
elif left_bc is not None and bc_id[:4] == "left":
bcs.append(
DirichletBC(V, Constant(left_bc), boundary,
topology_info[bc_id]))
elif top_bc is not None and bc_id[:3] == "top":
bcs.append(
DirichletBC(V, Constant(top_bc), boundary,
topology_info[bc_id]))
elif right_bc is not None and bc_id[:5] == "right":
bcs.append(
DirichletBC(V, Constant(right_bc), boundary,
topology_info[bc_id]))
else:
print(bc_id + " Not assigned as boundary condition ")
# raise NotImplementedError
# Define new measures associated with the interior domains and
# exterior boundaries
dx = Measure("dx", subdomain_data=subdomains)
ds = Measure("ds", subdomain_data=boundary)
f = Constant(0)
g = Constant(0)
if geometry is not None: # run multipatch implementation (Multiple domains)
a = []
L = []
for patch_id in geometry.keys():
kappa = geometry[patch_id].get("kappa")
a.append(
inner(Constant(kappa) * nabla_grad(u), nabla_grad(v)) *
dx(topology_info[patch_id]))
L.append(f * v * dx(topology_info[patch_id]))
a = sum(a)
L = sum(L)
else:
a = inner(Constant(kappa) * nabla_grad(u), nabla_grad(v)) * dx
L = f * v * dx
## Redefine u as a function in function space V for the solution
u = Function(V)
# Solve
solve(a == L, u, bcs)
u.rename('u', 'Temperature')
# Save solution to file in VTK format
print(' [+] Output to %s_solution.pvd' % filepath.split('.')[0])
vtkfile = File('%s_solution.pvd' % filepath.split('.')[0])
vtkfile << u
if show:
import matplotlib
matplotlib.use("Qt5Agg")
# Plot solution and gradient
plot(u, title="Temperature")
plt.gca().view_init(azim=-90, elev=90)
plt.show()
dofs = V.tabulate_dof_coordinates().reshape(
V.dim(),
mesh.geometry().dim()) #coordinates of nodes
vals = u.vector().get_local() #temperature at nodes
if save_solution:
from dolfin import HDF5File, MPI
output_file = HDF5File(MPI.comm_world,
filepath.split('.')[0] + "_solution_field.h5",
"w")
output_file.write(u, "solution")
output_file.close()
u.set_allow_extrapolation(True)
return dofs, vals, mesh, u
