def test_pickling():
# some simple checks for pickle
mesh = MeshQuad()
elem = ElementQuad1()
mapping = MappingIsoparametric(mesh, elem)
basis = CellBasis(mesh, elem, mapping)
pickled_mesh = pickle.dumps(mesh)
pickled_elem = pickle.dumps(elem)
pickled_mapping = pickle.dumps(mapping)
pickled_basis = pickle.dumps(basis)
mesh1 = pickle.loads(pickled_mesh)
elem1 = pickle.loads(pickled_elem)
mapping1 = pickle.loads(pickled_mapping)
basis1 = pickle.loads(pickled_basis)
assert_almost_equal(
laplace.assemble(basis).toarray(),
laplace.assemble(basis1).toarray(),
)
assert_almost_equal(
mesh.doflocs,
mesh1.doflocs,
)
assert_almost_equal(
mapping.J(0, 0, np.array([[.3], [.3]])),
mapping1.J(0, 0, np.array([[.3], [.3]])),
)
assert_almost_equal(
elem.doflocs,
elem1.doflocs,
)
def runTest(self):
m = self.mesh().refined(4)
basis = InteriorBasis(m, self.elem)
boundary_basis = FacetBasis(m, self.elem)
boundary_dofs = boundary_basis.get_dofs().flatten()
def dirichlet(x):
"""return a harmonic function"""
return ((x[0] + 1.j * x[1])**2).real
u = basis.zeros()
A = laplace.assemble(basis)
u[boundary_dofs] = projection(dirichlet,
boundary_basis,
I=boundary_dofs)
u = solve(*enforce(A, x=u, D=boundary_dofs))
@Functional
def gradu(w):
gradu = w['sol'].grad
return dot(gradu, gradu)
self.assertAlmostEqual(
gradu.assemble(basis, sol=basis.interpolate(u)),
8 / 3,
delta=1e-10,
)
def test_periodic_loading(m, mdgtype, e):
def _sort(ix):
# sort index arrays so that ix[0] matches
return ix[np.argsort(np.sum(m.p, axis=0)[ix])]
mp = mdgtype.periodic(m, _sort(m.nodes_satisfying(lambda x: x[0] == 0)),
_sort(m.nodes_satisfying(lambda x: x[0] == 1)))
basis = Basis(mp, e)
A = laplace.assemble(basis)
M = mass.assemble(basis)
@LinearForm
def linf(v, w):
return np.sin(2. * np.pi * w.x[0]) * v
f = linf.assemble(basis)
x = solve(A + 1e-6 * M, f)
def uexact(x):
return (1. / (2. * np.pi)**2) * np.sin(2. * np.pi * x[0])
@Functional
def func(w):
uexact = (1. / (2. * np.pi)**2) * np.sin(2. * np.pi * w.x[0])
return (uexact - w['u'])**2
assert_almost_equal(func.assemble(basis, u=x), 0, decimal=5)
def test_solving_inhomogeneous_laplace(mesh_elem, impose):
"""Adapted from example 14."""
mesh, elem = mesh_elem
m = mesh().refined(4)
basis = Basis(m, elem)
boundary_basis = FacetBasis(m, elem)
boundary_dofs = boundary_basis.get_dofs().flatten()
def dirichlet(x):
"""return a harmonic function"""
return ((x[0] + 1.j * x[1])**2).real
u = basis.zeros()
A = laplace.assemble(basis)
u[boundary_dofs] = projection(dirichlet, boundary_basis, I=boundary_dofs)
u = solve(*impose(A, x=u, D=boundary_dofs))
@Functional
def gradu(w):
gradu = w['sol'].grad
return dot(gradu, gradu)
np.testing.assert_almost_equal(gradu.assemble(basis,
sol=basis.interpolate(u)),
8 / 3,
decimal=9)
def test_subdomain_facet_assembly():
def subdomain(x):
return np.logical_and(
np.logical_and(x[0] > .25, x[0] < .75),
np.logical_and(x[1] > .25, x[1] < .75),
)
m, e = MeshTri().refined(4), ElementTriP2()
cbasis = CellBasis(m, e)
cbasis_p0 = cbasis.with_element(ElementTriP0())
sfbasis = FacetBasis(m, e, facets=m.facets_around(subdomain, flip=True))
sfbasis_p0 = sfbasis.with_element(ElementTriP0())
sigma = cbasis_p0.zeros() + 1
@BilinearForm
def laplace(u, v, w):
return dot(w.sigma * grad(u), grad(v))
A = laplace.assemble(cbasis, sigma=cbasis_p0.interpolate(sigma))
u0 = cbasis.zeros()
u0[cbasis.get_dofs(elements=subdomain)] = 1
u0_dofs = cbasis.get_dofs() + cbasis.get_dofs(elements=subdomain)
A, b = enforce(A, D=u0_dofs, x=u0)
u = solve(A, b)
@Functional
def measure_current(w):
return dot(w.n, w.sigma * grad(w.u))
meas = measure_current.assemble(sfbasis,
sigma=sfbasis_p0.interpolate(sigma),
u=sfbasis.interpolate(u))
assert_almost_equal(meas, 9.751915526759191)
def runTest(self):
m = self.init_mesh()
basis = InteriorBasis(m, self.element_type())
A = laplace.assemble(basis)
b = unit_load.assemble(basis)
x = solve(*condense(A, b, D=basis.get_dofs()))
self.assertAlmostEqual(np.max(x), self.maxval, places=3)
def runTest(self):
path = Path(__file__).parents[1] / 'docs' / 'examples' / 'meshes'
m = self.mesh_type.load(path / self.filename)
basis = InteriorBasis(m, self.element_type())
A = laplace.assemble(basis)
b = unit_load.assemble(basis)
x = solve(*condense(A, b, D=basis.get_dofs()))
self.assertAlmostEqual(np.max(x), 0.06261690318912218, places=3)
def test_periodic_mesh_assembly(m, mdgtype, etype, check1, check2):
def _sort(ix):
# sort index arrays so that ix[0] matches
return ix[np.argsort(np.sum(m.p, axis=0)[ix])]
mp = mdgtype.periodic(m, _sort(m.nodes_satisfying(check1)),
_sort(m.nodes_satisfying(check2)))
basis = Basis(mp, etype())
A = laplace.assemble(basis)
f = unit_load.assemble(basis)
D = basis.get_dofs()
x = solve(*condense(A, f, D=D))
if m.dim() == 2:
assert_almost_equal(x.max(), 0.125)
else:
assert_almost_equal(x.max(), 0.07389930610869411, decimal=3)
assert not np.isnan(x).any()
def laplace(m, **params):
"""Solve the Laplace equation using the FEM.
Parameters
----------
m
A Mesh object.
"""
e = ElementTriP1()
basis = CellBasis(m, e)
A = laplacian.assemble(basis)
b = unit_load.assemble(basis)
u = solve(*condense(A, b, I=m.interior_nodes()))
# evaluate the error estimators
fbasis = [InteriorFacetBasis(m, e, side=i) for i in [0, 1]]
w = {"u" + str(i + 1): fbasis[i].interpolate(u) for i in [0, 1]}
@Functional
def interior_residual(w):
h = w.h
return h**2
eta_K = interior_residual.elemental(basis)
@Functional
def edge_jump(w):
h = w.h
n = w.n
dw1 = grad(w["u1"])
dw2 = grad(w["u2"])
return h * ((dw1[0] - dw2[0]) * n[0] + (dw1[1] - dw2[1]) * n[1])**2
eta_E = edge_jump.elemental(fbasis[0], **w)
tmp = np.zeros(m.facets.shape[1])
np.add.at(tmp, fbasis[0].find, eta_E)
eta_E = np.sum(0.5 * tmp[m.t2f], axis=0)
return eta_E + eta_K
def assembler(m):
basis = Basis(m, ElementTetP1())
return (
laplace.assemble(basis),
unit_load.assemble(basis),
)