def runTest(self):
"""Solve Stokes problem, try splitting and other small things."""
m = MeshTri().refined()
m = m.refined(3).with_boundaries({
'up': lambda x: x[1] == 1.,
'rest': lambda x: x[1] != 1.,
})
e = ElementVectorH1(ElementTriP2()) * ElementTriP1()
basis = CellBasis(m, e)
@BilinearForm
def bilinf(u, p, v, q, w):
from skfem.helpers import grad, ddot, div
return (ddot(grad(u), grad(v)) - div(u) * q - div(v) * p
- 1e-2 * p * q)
S = asm(bilinf, basis)
D = basis.find_dofs(skip=['u^2'])
x = basis.zeros()
x[D['up'].all('u^1^1')] = .1
x = solve(*condense(S, x=x, D=D))
(u, u_basis), (p, p_basis) = basis.split(x)
self.assertEqual(len(u), m.p.shape[1] * 2 + m.facets.shape[1] * 2)
self.assertEqual(len(p), m.p.shape[1])
self.assertTrue(np.sum(p - x[basis.nodal_dofs[2]]) < 1e-8)
U, P = basis.interpolate(x)
self.assertTrue(isinstance(U.value, np.ndarray))
self.assertTrue(isinstance(P.value, np.ndarray))
self.assertTrue(P.shape[0] == m.nelements)
self.assertTrue((basis.doflocs[:, D['up'].all()][1] == 1.).all())
# test blocks splitting of forms while at it
C1 = asm(bilinf.block(1, 1), CellBasis(m, ElementTriP1()))
C2 = S[basis.nodal_dofs[-1]].T[basis.nodal_dofs[-1]].T
self.assertTrue(abs((C1 - C2).min()) < 1e-10)
self.assertTrue(abs((C1 - C2).max()) < 1e-10)
# test splitting ElementVector
(ux, uxbasis), (uy, uybasis) = u_basis.split(u)
assert_allclose(ux[uxbasis.nodal_dofs[0]], u[u_basis.nodal_dofs[0]])
assert_allclose(ux[uxbasis.facet_dofs[0]], u[u_basis.facet_dofs[0]])
assert_allclose(uy[uybasis.nodal_dofs[0]], u[u_basis.nodal_dofs[1]])
assert_allclose(uy[uybasis.facet_dofs[0]], u[u_basis.facet_dofs[1]])
def plot(self, *args, **kwargs):
"""Convenience wrapper for :func:`skfem.visuals.matplotlib.plot`."""
from skfem.visuals.matplotlib import plot, show
from skfem.assembly import CellBasis
ax = plot(CellBasis(self, self.elem()), *args, **kwargs)
ax.show = show
return ax
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 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 test_point_source(etype):
mesh = MeshLine1().refined()
basis = CellBasis(mesh, etype())
source = np.array([0.7])
u = solve(*condense(asm(laplace, basis), basis.point_source(source), D=basis.get_dofs()))
exact = np.stack([(1 - source) * mesh.p, (1 - mesh.p) * source]).min(0)
assert_almost_equal(u[basis.nodal_dofs], exact)
def test_basis_interpolate_project(m, e):
basis = CellBasis(m, e)
x = basis.zeros() + 1
Y = basis.interpolate(x)
X = basis.project(Y)
# check that interpolate and project are inverses
assert_almost_equal(x, X)
def test_oriented_gauss_integral(m, e):
facets = m.facets_around('mid')
fb = FacetBasis(m, e, facets=facets)
cb = CellBasis(m, e, elements='mid')
assert_almost_equal(
Functional(lambda w: w.x[0] * w.n[0]).assemble(fb),
Functional(lambda w: 1. + 0. * w.x[0]).assemble(cb),
decimal=5,
)
def test_basis_project_grad():
m, e = (MeshTri(), ElementTriP1())
basis = CellBasis(m, e)
Y = basis.interpolate(m.p[0])
y = basis.project(Y.grad[0])
@Functional
def int_y(w):
return w.y ** 2
assert_almost_equal(int_y.assemble(basis, y=y), 1.)
def runTest(self):
m = self.prepare_mesh()
basis = CellBasis(m, self.element_type())
x = projection(lambda x: x[0] ** 2, basis)
fun = basis.interpolator(x)
X = np.linspace(0, 1, 10)
dim = m.dim()
if dim == 3:
y = fun(np.array([X, [0.31] * 10, [0.62] * 10]))
elif dim == 2:
y = fun(np.array([X, [0.31] * 10]))
else:
y = fun(np.array([X]))
assert_allclose(y, X ** 2, atol=1e-10)
def test_basis_project(m, e, fun):
basis = CellBasis(m, e)
y = basis.project(fun)
@Functional
def int_y_1(w):
return w.y**2
@Functional
def int_y_2(w):
return fun(w.x)**2
assert_almost_equal(int_y_1.assemble(basis, y=y), int_y_2.assemble(basis))
def runTest(self):
mesh = MeshTri().refined(5)
basis = CellBasis(mesh, ElementTriP1())
def fun(X):
x, y = X
return x**2 + y**2
x = projection(fun, basis)
y = fun(mesh.p)
normest = np.linalg.norm(x - y)
self.assertTrue(normest < 0.011, msg="|x-y| = {}".format(normest))
def runTest(self):
m = self.mesh_type()
basis = CellBasis(m, self.mesh_type.elem())
y = basis.project(lambda x: x[0])
plot(basis,
y,
nrefs=1,
figsize=(4, 4),
aspect=1.1,
cmap=plt.get_cmap('viridis'),
shading='gouraud',
vmin=0,
vmax=1,
levels=3,
colorbar=True)
def test_trace(mtype, e1, e2, flat):
m = mtype().refined(3)
# use the boundary where last coordinate is zero
basis = FacetBasis(m, e1, facets=m.facets_satisfying(lambda x: x[-1] == 0.0))
xfun = projection(lambda x: x[0], CellBasis(m, e1))
nbasis, y = basis.trace(xfun, lambda p: p[0] if flat else p[:-1], target_elem=e2)
@Functional
def integ(w):
return w.y
# integrate f(x) = x_1 over trace mesh
assert_almost_equal(integ.assemble(nbasis, y=nbasis.interpolate(y)), .5)
def test_interpolator_probes(mtype, e, nrefs, npoints):
m = mtype()
if nrefs > 0:
m = m.refined(nrefs)
np.random.seed(0)
X = np.random.rand(m.p.shape[0], int(npoints))
basis = CellBasis(m, e)
y = projection(lambda x: x[0] + x[1], basis)
assert_allclose(basis.probes(X) @ y, basis.interpolator(y)(X))
assert_allclose(basis.probes(X) @ y, X[0] + X[1])
def runTest(self):
m = self.mesh_type().refined(2)
basis = CellBasis(m, self.elem_type())
for fun in [
lambda x: x[0] == 0, lambda x: x[0] == 1, lambda x: x[1] == 0,
lambda x: x[1] == 1, lambda x: x[2] == 0, lambda x: x[2] == 1
]:
arr1 = basis.find_dofs({'kek':
m.facets_satisfying(fun)})['kek'].edge['u']
arr2 = basis.edge_dofs[:, m.edges_satisfying(fun)]
assert_allclose(arr1, arr2.flatten())
def runTest(self):
m = self.mesh
e = ElementTriP1()
basis = CellBasis(m, e)
A = laplace.assemble(basis)
M = mass.assemble(basis)
D = m.boundary_nodes()
assert_almost_equal(enforce(A, D=D).toarray(), np.eye(A.shape[0]))
assert_almost_equal(
enforce(M, D=D, diag=0.).toarray(), np.zeros(M.shape))
enforce(A, D=D, overwrite=True)
assert_almost_equal(A.toarray(), np.eye(A.shape[0]))
def test_simple_cg_solver():
m = MeshTri().refined(3)
basis = CellBasis(m, ElementTriP1())
A0 = laplace.coo_data(basis)
A1 = laplace.assemble(basis)
f = unit_load.assemble(basis)
D = m.boundary_nodes()
x1 = solve(*condense(A1, f, D=D))
f[D] = 0
x0 = A0.solve(f, D=D)
assert_almost_equal(x0, x1)
def runTest(self):
"""Solve Stokes problem, try splitting and other small things."""
m = MeshTri().refined()
m = m.refined(3).with_boundaries({
'up': lambda x: x[1] == 1.,
'rest': lambda x: x[1] != 1.,
})
e = ElementVectorH1(ElementTriP2()) * ElementTriP1()
basis = CellBasis(m, e)
@BilinearForm
def bilinf(u, p, v, q, w):
from skfem.helpers import grad, ddot, div
return (ddot(grad(u), grad(v)) - div(u) * q - div(v) * p
- 1e-2 * p * q)
S = asm(bilinf, basis)
D = basis.find_dofs(skip=['u^2'])
x = basis.zeros()
x[D['up'].all('u^1^1')] = .1
x = solve(*condense(S, x=x, D=D))
(u, u_basis), (p, p_basis) = basis.split(x)
self.assertEqual(len(u), m.p.shape[1] * 2 + m.facets.shape[1] * 2)
self.assertEqual(len(p), m.p.shape[1])
self.assertTrue(np.sum(p - x[basis.nodal_dofs[2]]) < 1e-8)
U, P = basis.interpolate(x)
self.assertTrue(isinstance(U.value, np.ndarray))
self.assertTrue(isinstance(P.value, np.ndarray))
self.assertTrue((basis.doflocs[:, D['up'].all()][1] == 1.).all())
def test_call_svg_plot(mtype):
m = mtype()
svg = drawsvg(m, nrefs=2)
basis = CellBasis(m, mtype.elem())
svg_plot = plotsvg(basis, m.p[0], nrefs=2)
def runTest(self):
with self.assertRaises(ValueError):
m = MeshTri()
e = ElementTetP2()
basis = CellBasis(m, e)
def test_basis_project_composite(m, e, p1, p2):
basis = CellBasis(m, e)
y = basis.project(p1)
assert_almost_equal(y, p2)
