def flip_vectors(mesh, eigs, mesh_ref, eigs_ref, test_it=False):
x_c = 1e-3
for i in range(len(eigs)):
s = FESolution(mesh, eigs[i])
s_ref = FESolution(mesh_ref, eigs_ref[i])
if s.value(x_c) < 0:
#print " Multiplying %d-th coarse eigenvector by (-1)" % i
eigs[i] = -eigs[i]
if s_ref.value(x_c) < 0:
#print " Multiplying %d-th ref. eigenvector by (-1)" % i
eigs_ref[i] = -eigs_ref[i]
if test_it:
# Test it:
s = FESolution(mesh, eigs[i]).to_discrete_function()
s_ref = FESolution(mesh_ref, eigs_ref[i]).to_discrete_function()
same_norm = (s - s_ref).l2_norm()
flipped_norm = (s + s_ref).l2_norm()
print same_norm, flipped_norm
if same_norm > flipped_norm:
c = min(same_norm, flipped_norm) / max(same_norm, flipped_norm)
print "Warning: the flip is wrong, c=", c
# If "c" is almost one, then the vectors can't really be
# aligned anyway:
assert c > 0.9
def project_onto(self, mesh, proj_type="Fekete"):
"""
Projects 'self' onto the 'mesh' using the 'proj_type' projection.
proj_type == "Fekete"/"L2"/"H1"
"""
if mesh == self._mesh:
return self
if proj_type == "Fekete":
return Function(self, mesh)
elif proj_type in ["L2", "H1"]:
from hermes1d.h1d_wrapper.h1d_wrapper import \
(assemble_projection_matrix_rhs, Mesh, FESolution)
from hermes1d.hermes_common.matrix import CSCMatrix, AVector
pts, orders = mesh.get_mesh_data()
m = Mesh(pts, orders)
n_dof = m.assign_dofs()
A = CSCMatrix(n_dof)
rhs = AVector(n_dof)
assemble_projection_matrix_rhs(m,
A,
rhs,
self,
projection_type=proj_type)
coeffs = solve(A.to_scipy_csc().todense(), rhs.to_numpy())
return FESolution(m, coeffs).to_discrete_function()
else:
raise ValueError("Unknown projection type")
def test_l2_h1_proj_run():
"""
Test that the projections run.
It doesn't test if it's correct.
"""
pts = arange(0, 2 * pi, 1)
orders = [3] * (len(pts) - 1)
m = Mesh(pts, orders)
n_dof = m.assign_dofs()
A = CSCMatrix(n_dof)
rhs = AVector(n_dof)
assemble_projection_matrix_rhs(m, A, rhs, f_sin, projection_type="L2")
x = solve(A.to_scipy_csc().todense(), rhs.to_numpy())
sol_l2 = FESolution(m, x).to_discrete_function()
A = CSCMatrix(n_dof)
assemble_projection_matrix_rhs(m, A, rhs, f_sin, projection_type="H1")
x = solve(A.to_scipy_csc().todense(), rhs.to_numpy())
sol_h1 = FESolution(m, x).to_discrete_function()
sol_l2.plot(False)
sol_h1.plot(False)
def test_l2_h1_proj3():
"""
Tests conversion to FE basis.
"""
pts = arange(0, 2 * pi, 0.1)
orders = [2] * (len(pts) - 1)
m = Mesh(pts, orders)
f = Function(lambda x: sin(x), Mesh1D(pts, orders))
n_dof = m.assign_dofs()
A = CSCMatrix(n_dof)
rhs = AVector(n_dof)
assemble_projection_matrix_rhs(m, A, rhs, f, projection_type="L2")
x = solve(A.to_scipy_csc().todense(), rhs.to_numpy())
sol_l2 = FESolution(m, x).to_discrete_function()
A = CSCMatrix(n_dof)
assemble_projection_matrix_rhs(m, A, rhs, f, projection_type="H1")
x = solve(A.to_scipy_csc().todense(), rhs.to_numpy())
sol_h1 = FESolution(m, x).to_discrete_function()
assert sol_l2 == f
assert sol_h1 == f
def calculate_FE_coeffs(self):
if self._fe_sol is None:
from hermes1d.h1d_wrapper.h1d_wrapper import \
(assemble_projection_matrix_rhs, Mesh, FESolution)
from hermes_common._hermes_common import CooMatrix
pts, orders = self._mesh.get_mesh_data()
m = Mesh(pts, orders)
n_dof = m.assign_dofs()
A = CooMatrix(n_dof)
rhs = empty(n_dof)
assemble_projection_matrix_rhs(m, A, rhs, self,
projection_type="L2")
coeffs = solve(A.to_scipy_coo().todense(), rhs)
self._fe_sol = FESolution(m, coeffs)
def test_l2_h1_proj2():
"""
Tests the correctness of the projections.
"""
pts = arange(0, 2 * pi, 3)
orders = [4] * (len(pts) - 1)
m = Mesh(pts, orders)
pts = array(list(arange(0, pts[-1], 0.1)) + [pts[-1]])
orders = [6] * (len(pts) - 1)
f_exact = Function(lambda x: sin(x), Mesh1D(pts, orders))
n_dof = m.assign_dofs()
A = CSCMatrix(n_dof)
rhs = AVector(n_dof)
assemble_projection_matrix_rhs(m, A, rhs, f_sin, projection_type="L2")
x = solve(A.to_scipy_csc().todense(), rhs.to_numpy())
sol_l2 = FESolution(m, x).to_discrete_function()
A = CSCMatrix(n_dof)
assemble_projection_matrix_rhs(m, A, rhs, f_sin, projection_type="H1")
x = solve(A.to_scipy_csc().todense(), rhs.to_numpy())
sol_h1 = FESolution(m, x).to_discrete_function()
assert (sol_l2 - f_exact).l2_norm() < 0.002
assert (sol_h1 - f_exact).l2_norm() < 0.002
def refine_mesh_romanowski(mesh, solutions):
"""
Uses Romanowski refinement for all solutions in 'solutions'.
Solutions are given as vectors coming from the matrix solver.
"""
els2refine = []
errors = []
for sol in solutions:
s = FESolution(mesh, sol)
id, error = find_element_romanowski(s.get_element_coeffs())
els2refine.append(id)
errors.append(error)
els2refine = list(set(els2refine))
print "Will refine the elements:", els2refine
mesh = refine_mesh(mesh, els2refine)
return mesh
def adapt_mesh(mesh, eigs, l=0, Z=1, adapt_type="hp", eqn_type="R"):
"""
Adapts the mesh using the adaptivity type 'adapt_type'.
Returns a new instance of the H1D mesh.
adapt_type .... one of: h, hp, p, uniform-p, romanowski
"""
if adapt_type == "romanowski":
m = refine_mesh_romanowski(mesh, eigs)
pts, orders = m.get_mesh_data()
return Mesh(pts, orders)
elif adapt_type == "uniform-p":
pts, orders = mesh.get_mesh_data()
orders = array(orders) + 1
return Mesh(pts, orders)
elif adapt_type in ["h", "p", "hp"]:
NORM = 1 # 1 ... H1; 0 ... L2;
THRESHOLD = 0.7
mesh_ref = mesh.reference_refinement()
print "Fine mesh created (%d DOF)." % mesh_ref.get_n_dof()
N_dof, energies, eigs_ref = solve_schroedinger(mesh_ref,
l=l,
Z=Z,
eqn_type=eqn_type,
eig_num=len(eigs))
flip_vectors(mesh, eigs, mesh_ref, eigs_ref)
print " Done."
sols = []
sols_ref = []
print "Normalizing solutions..."
for i in range(len(eigs)):
e = (eigs[i]).copy()
coarse_h1_norm = FESolution(mesh, e).h1_norm()
e /= coarse_h1_norm
sols.append(e)
e = (eigs_ref[i]).copy()
reference_h1_norm = FESolution(mesh_ref, e).h1_norm()
e /= reference_h1_norm
sols_ref.append(e)
#print "H1 norms:"
#print "coarse (%d):" % i, coarse_h1_norm
#print "reference (%d):" % i, reference_h1_norm
print " Done."
meshes = []
mesh_orig = mesh.copy()
mesh_orig.assign_dofs()
errors = []
for sol, sol_ref in zip(sols, sols_ref):
mesh = mesh_orig.copy()
mesh.assign_dofs()
mesh_ref = mesh.reference_refinement()
mesh_ref.assign_dofs()
mesh.copy_vector_to_mesh(sol, 0)
mesh_ref.copy_vector_to_mesh(sol_ref, 0)
err_est_total, err_est_array = calc_error_estimate(
NORM, mesh, mesh_ref)
ref_sol_norm = calc_solution_norm(NORM, mesh_ref)
err_est_rel = err_est_total / ref_sol_norm
print "Relative error (est) = %g %%\n" % (100. * err_est_rel)
errors.append(err_est_rel)
# TODO: adapt using all the vectors:
# 0 ... hp, 1 ... h, 2 ... p
if adapt_type == "hp":
ADAPT_TYPE = 0
elif adapt_type == "h":
ADAPT_TYPE = 1
elif adapt_type == "p":
ADAPT_TYPE = 2
else:
raise ValueError("Unkown adapt_type")
adapt(NORM, ADAPT_TYPE, THRESHOLD, err_est_array, mesh, mesh_ref)
meshes.append(mesh)
pts, orders = mesh_orig.get_mesh_data()
mesh = Mesh1D(pts, orders)
for m in meshes:
pts, orders = m.get_mesh_data()
m = Mesh1D(pts, orders)
mesh = mesh.union(m)
pts, orders = mesh.get_mesh_data()
mesh = Mesh(pts, orders)
return mesh
else:
raise ValueError("Unknown adapt_type")
