class Line1D(TestCase):
"""Solve the following problem:
u'' = 0
u(0) = 0
u'(1) = 1
Solution is u(x) = x.
"""
e = ElementLineP1()
def runTest(self):
m = MeshLine(np.linspace(0., 1.)).refined(2)
ib = InteriorBasis(m, self.e)
fb = FacetBasis(m, self.e)
@LinearForm
def boundary_flux(v, w):
return v * (w.x[0] == 1.)
L = asm(laplace, ib)
b = asm(boundary_flux, fb)
D = m.nodes_satisfying(lambda x: x == 0.0)
I = ib.complement_dofs(D) # noqa E741
u = solve(*condense(L, b, I=I)) # noqa E741
np.testing.assert_array_almost_equal(u[ib.nodal_dofs[0]], m.p[0], -10)
class LineNegative1D(TestCase):
"""Solve the following problem:
u'' = 0
u(0) = 0
u'(1) = -1
Solution is u(x) = -x.
"""
e = ElementLineP1()
def runTest(self):
m = MeshLine(np.linspace(0., 1.)).refined(2).with_boundaries({
'left':
lambda x: x[0] == 0.0,
'right':
lambda x: x[0] == 1.0,
})
ib = InteriorBasis(m, self.e)
fb = FacetBasis(m, self.e, facets=m.boundaries['right'])
@LinearForm
def boundary_flux(v, w):
return -w.x[0] * v
L = asm(laplace, ib)
b = asm(boundary_flux, fb)
D = ib.find_dofs()['left'].all()
I = ib.complement_dofs(D) # noqa E741
u = solve(*condense(L, b, I=I)) # noqa E741
np.testing.assert_array_almost_equal(u[ib.nodal_dofs[0]], -m.p[0], -10)
class TestPartitionofUnity(TestCase):
"""Test that elements form a partition of unity."""
elems = [
ElementLineP1(),
ElementLineP2(),
ElementTriP1(),
ElementTriP2(),
ElementQuad1(),
ElementQuad2(),
ElementQuadS2(),
ElementTetP1(),
ElementTetP2(),
ElementHex1(),
ElementHexS2(),
ElementHex2(),
]
def runTest(self):
for elem in self.elems:
if elem.dim == 1:
y = np.array([[.15]])
elif elem.dim == 2:
y = np.array([[.15], [.15]])
elif elem.dim == 3:
y = np.array([[.15], [.15], [.15]])
out = 0.
for i in range(elem.doflocs.shape[0]):
out += elem.lbasis(y, i)[0][0]
self.assertAlmostEqual(out, 1, msg='failed for {}'.format(elem))
class LineNeumann1D(TestCase):
"""Solve the following problem:
-u'' + eps*u = 0
u'(0) = 1
u'(1) = 1
Solution is u(x) = x-0.5.
"""
e = ElementLineP1()
def runTest(self):
m = MeshLine(np.linspace(0., 1.)).refined(2)
ib = InteriorBasis(m, self.e)
fb = FacetBasis(m, self.e)
@LinearForm
def boundary_flux(v, w):
return v * (w.x[0] == 1) - v * (w.x[0] == 0)
L = asm(laplace, ib)
M = asm(mass, ib)
b = asm(boundary_flux, fb)
u = solve(L + 1e-6 * M, b)
np.testing.assert_array_almost_equal(u[ib.nodal_dofs[0]], m.p[0] - .5,
-4)
class TestNodality(TestCase):
"""Test for Element.doflocs."""
elems = [
ElementLineP0(),
ElementLineP1(),
ElementLineP2(),
ElementLinePp(1),
ElementLinePp(3),
ElementLineMini(),
ElementTriP0(),
ElementTriP1(),
ElementTriP2(),
ElementTriP3(),
ElementTriP4(),
ElementTriMini(),
ElementQuad0(),
ElementQuad1(),
ElementQuad2(),
ElementQuadS2(),
ElementQuadP(1),
ElementQuadP(3),
ElementTetP0(),
ElementTetP1(),
ElementTetP2(),
ElementTetMini(),
ElementHex1(),
ElementHexS2(),
ElementHex2(),
ElementTetCR(),
ElementTetCCR(),
ElementTriCR(),
ElementTriCCR(),
ElementWedge1(),
]
def runTest(self):
for e in self.elems:
N = e.doflocs.shape[0]
Ih = np.zeros((N, N))
for itr in range(N):
Ih[itr] = e.lbasis(e.doflocs.T, itr)[0]
# Remove nan-rows: test nodality only on non-nan doflocs.
#
# Some elements, such as ElementTriMini might have a combination
# of nodal dofs and non-nodal dofs.
#
# Nodal dof is defined so that there exists a point where the
# corresponding basis function is one, and other basis functions
# are zero. Non-nodal dof does not satisfy this property.
ix = np.isnan(np.sum(Ih, axis=1))
Nnan = np.sum(ix)
ixs = np.nonzero(~ix)[0]
Ih = Ih[ixs].T[ixs].T
assert_allclose(Ih,
np.eye(N - Nnan),
atol=1e-13,
err_msg="{}".format(type(e)))
class TestDerivatives(TestCase):
"""Test values of derivatives."""
elems = [
ElementLineP1(),
ElementLineP2(),
ElementTriP1(),
ElementTriP2(),
ElementTriMini(),
ElementQuad1(),
ElementQuad2(),
ElementQuadS2(),
ElementTetP1(),
ElementTetP2(),
ElementTetMini(),
ElementHex1(),
ElementHexS2(),
]
def runTest(self):
for elem in self.elems:
eps = 1e-6
for base in [0., .3, .6, .9]:
if elem.dim == 1:
y = np.array([[base, base + eps]])
elif elem.dim == 2:
y = np.array([[base, base + eps, base, base],
[base, base, base, base + eps]])
elif elem.dim == 3:
y = np.array([[base, base + eps, base, base, base, base],
[base, base, base, base + eps, base, base],
[base, base, base, base, base, base + eps]])
i = 0
while True:
try:
out = elem.lbasis(y, i)
except ValueError:
break
diff = (out[0][1] - out[0][0]) / eps
errmsg = 'x-derivative for {}th bfun failed for {}'
self.assertAlmostEqual(diff,
out[1][0][0],
delta=1e-3,
msg=errmsg.format(i, elem))
if elem.dim > 1:
diff = (out[0][3] - out[0][2]) / eps
errmsg = 'y-derivative for {}th bfun failed for {}'
self.assertAlmostEqual(diff,
out[1][1][3],
delta=1e-3,
msg=errmsg.format(i, elem))
if elem.dim == 3:
diff = (out[0][5] - out[0][4]) / eps
errmsg = 'z-derivative for {}th bfun failed for {}'
self.assertAlmostEqual(diff,
out[1][2][4],
delta=1e-3,
msg=errmsg.format(i, elem))
i += 1
def runTest(self):
m = MeshLine(np.linspace(0., 1.))
m.refine(2)
e = ElementLineP1()
ib = InteriorBasis(m, e)
fb = FacetBasis(m, e)
@linear_form
def boundary_flux(v, dv, w):
return v * (w.x[0] == 1) - v * (w.x[0] == 0)
L = asm(laplace, ib)
M = asm(mass, ib)
b = asm(boundary_flux, fb)
u = solve(L + 1e-6 * M, b)
self.assertTrue(np.sum(np.abs(u - m.p[0, :] + 0.5)) < 1e-4)
def runTest(self):
m = MeshLine(np.linspace(0., 1.))
m.refine(2)
e = ElementLineP1()
ib = InteriorBasis(m, e)
fb = FacetBasis(m, e)
@linear_form
def boundary_flux(v, dv, w):
return -v * (w.x[0] == 1.)
L = asm(laplace, ib)
b = asm(boundary_flux, fb)
D = m.nodes_satisfying(lambda x: x == 0.0)
I = ib.complement_dofs(D) # noqa E741
u = solve(*condense(L, b, I=I)) # noqa E741
self.assertTrue(np.sum(np.abs(u + m.p[0, :])) < 1e-10)
def mapping(self):
return MappingIsoparametric(self, ElementQuad1(), ElementLineP1())
ElementQuad1(),
),
(
MeshQuad().refined(2),
MeshQuad1DG,
ElementQuad2(),
),
(
MeshTri().refined(2),
MeshTri1DG,
ElementTriP2(),
),
(
MeshLine().refined(5),
MeshLine1DG,
ElementLineP1(),
),
(
MeshHex().refined(3),
MeshHex1DG,
ElementHex1(),
),
(
MeshLine().refined(),
MeshLine1DG,
ElementLinePp(5),
),
])
def test_periodic_loading(m, mdgtype, e):
def _sort(ix):
# sort index arrays so that ix[0] matches
def create_basis(self, m):
e = ElementLineP1()
return Basis(m, e)
def _mapping(self):
from skfem.mapping import MappingIsoparametric
from skfem.element import ElementQuad1, ElementLineP1
return MappingIsoparametric(self, ElementQuad1(), ElementLineP1())
element_type = ElementLineP2
nrefs = 5
class TestIncompatibleMeshElement(TestCase):
def runTest(self):
with self.assertRaises(ValueError):
m = MeshTri()
e = ElementTetP2()
basis = InteriorBasis(m, e)
@pytest.mark.parametrize("mtype,e1,e2", [
(MeshTri, ElementTriP1(), ElementLineP0()),
(MeshTri, ElementTriP1(), ElementLineP1()),
(MeshTri, ElementTriP2(), ElementLineP1()),
(MeshTri, ElementTriP2(), ElementLineP2()),
(MeshTri, ElementTriP2(), None),
(MeshQuad, ElementQuad1(), ElementLineP0()),
(MeshQuad, ElementQuad1(), ElementLineP1()),
(MeshQuad, ElementQuad2(), ElementLineP2()),
(MeshTet, ElementTetP1(), ElementTriP0()),
(MeshTet, ElementTetP2(), ElementTriP2()),
(MeshHex, ElementHex1(), ElementQuad0()),
(MeshHex, ElementHex1(), ElementQuad1()),
(MeshHex, ElementHex2(), ElementQuad2()),
])
def test_trace(mtype, e1, e2):
m = mtype().refined(3)