def __init__(self, doflocs, t, **kwargs):
warnings.warn("MeshQuad2 is an experimental feature and "
"not governed by the semantic versioning. "
"Several features of MeshQuad are still "
"missing.")
if t.shape[0] == 9:
dofs, ix = np.unique(t[:4], return_inverse=True)
super(MeshQuad2, self).__init__(
doflocs[:, dofs],
np.arange(len(dofs), dtype=np.int)[ix].reshape(t[:4].shape),
**kwargs)
else:
# fallback for refinterp
super(MeshQuad2, self).__init__(doflocs, t, **kwargs)
from skfem.element import ElementQuad1, ElementQuad2
from skfem.assembly import InteriorBasis
from skfem.mapping import MappingIsoparametric
self._elem = ElementQuad2()
self._basis = InteriorBasis(self, self._elem,
MappingIsoparametric(self, ElementQuad1()))
self._mesh = MeshQuad.from_basis(self._basis)
if t.shape[0] == 9:
self._mesh.p = doflocs
self._mesh.t = t
def __init__(self, doflocs, t, **kwargs):
warnings.warn("MeshTri2 is an experimental feature and "
"not governed by the semantic versioning. "
"Several features of MeshTri are still "
"missing.")
if t.shape[0] == 6:
dofs, ix = np.unique(t[:3], return_inverse=True)
super(MeshTri2, self).__init__(
doflocs[:, dofs],
np.arange(len(dofs), dtype=np.int)[ix].reshape(t[:3].shape),
sort_t=False,
**kwargs)
else:
# fallback for refinterp
super(MeshTri2, self).__init__(doflocs, t, **kwargs)
from skfem.element import ElementTriP2
from skfem.assembly import InteriorBasis
from skfem.mapping import MappingAffine
self._elem = ElementTriP2()
self._basis = InteriorBasis(self, self._elem, MappingAffine(self))
self._mesh = MeshTri.from_basis(self._basis)
if t.shape[0] == 6:
self._mesh.p = doflocs
self._mesh.t = t
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_dg_element(m, e, edg):
edg = edg(e)
@Functional
def square(w):
return w['random']**2
basis = InteriorBasis(m, e)
basisdg = InteriorBasis(m, edg)
assert_allclose(
square.assemble(basis, random=basis.interpolate(basis.zeros() + 1)),
square.assemble(basisdg,
random=basisdg.interpolate(basisdg.zeros() + 1)),
)
def runTest(self):
m = MeshTri()
basis = InteriorBasis(m, ElementTriP2())
dofs = basis.get_dofs()
self.assertEqual(len(dofs.nodal['u']), 4)
self.assertEqual(len(dofs.facet['u']), 4)
def runTest(self):
m = MeshQuad().refined(2)
e = ElementQuad1()
basis = InteriorBasis(m, e)
M, X = basis.refinterp(m.p[0], 3)
self.assertEqual(M.p.shape[1], len(X))
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):
m = MeshTri()
basis = InteriorBasis(m, ElementTriP2())
D1 = basis.get_dofs(lambda x: x[0] == 0)
D2 = basis.get_dofs(lambda x: x[0] == 1)
D3 = basis.get_dofs(lambda x: x[1] == 1)
D4 = basis.get_dofs(lambda x: x[1] == 0)
assert_allclose(D1 | D2 | D3 | D4, basis.get_dofs())
assert_allclose(D1 + D2 + D3 + D4, basis.get_dofs())
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 runTest(self):
mtype, etype = self.case
m = mtype().refined(3)
e = etype()
ib = InteriorBasis(m, e)
x = self.initOnes(ib)
f = ib.interpolator(x)
X = np.array([np.sin(m.p[0, :]), np.sin(3. * m.p[1, :])])
self.assertTrue(np.sum(f(X) - 1.0) < 1.0e-10)
def runTest(self):
mesh = MeshTri().refined(5)
basis = InteriorBasis(mesh, ElementTriP1())
def fun(x, y):
return x**2 + y**2
x = L2_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 = MeshQuad().refined(3)
e = ElementQuad1()
basis = InteriorBasis(m, e)
@Functional
def x_squared(w):
return w.x[0]**2
y = asm(x_squared, basis)
self.assertAlmostEqual(y, 1. / 3.)
self.assertEqual(len(x_squared.elemental(basis)), m.t.shape[1])
def runTest(self):
m = MeshHex()
# check that these assemble to the same matrix
ec = ElementHex1() * ElementHex1() * ElementHex1()
ev = ElementVectorH1(ElementHex1())
basisc = InteriorBasis(m, ec)
basisv = InteriorBasis(m, ev)
@BilinearForm
def bilinf_ev(u, v, w):
from skfem.helpers import dot
return dot(u, v)
@BilinearForm
def bilinf_ec(ux, uy, uz, vx, vy, vz, w):
return ux * vx + uy * vy + uz * vz
Kv = asm(bilinf_ev, basisv)
Kc = asm(bilinf_ec, basisc)
self.assertAlmostEqual(np.sum(np.sum((Kv - Kc).todense())), 0.)
def runTest(self):
m = self.mesh_type().refined(self.nrefs)
basis = InteriorBasis(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 runTest(self):
m = self.mesh_type().refined(2)
basis = InteriorBasis(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 test_evaluate_functional(mtype, e, mtype2):
m = mtype().refined(3)
if mtype2 is not None:
m = mtype2.from_mesh(m)
basis = InteriorBasis(m, e)
@Functional
def x_squared(w):
return w.x[0] ** 2
y = asm(x_squared, basis)
assert_almost_equal(y, 1. / 3.)
assert_equal(len(x_squared.elemental(basis)),
m.t.shape[1])
def runTest(self):
"""Solve Stokes problem, try splitting and other small things."""
m = MeshTri()
m.refine()
m.define_boundary('centreline',
lambda x: x[0] == .5,
boundaries_only=False)
m.refine(3)
e = ElementVectorH1(ElementTriP2()) * ElementTriP1()
m.define_boundary('up', lambda x: x[1] == 1.)
m.define_boundary('rest', lambda x: x[1] != 1.)
basis = InteriorBasis(m, e)
self.assertEqual(
basis.get_dofs(m.boundaries['centreline']).all().size,
(2 + 1) * (2**(1 + 3) + 1) + 2 * 2**(1 + 3))
self.assertEqual(basis.find_dofs()['centreline'].all().size,
(2 + 1) * (2**(1 + 3) + 1) + 2 * 2**(1 + 3))
@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, basis.zeros(), 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 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)
def runTest(self):
m = self.mesh().refined(3)
ib = InteriorBasis(m, self.elem())
A = asm(laplace, ib)
D = ib.get_dofs().all()
I = ib.complement_dofs(D)
for X in self.funs:
x = self.set_bc(X, ib)
Xh = x.copy()
x = solve(*condense(A, x=x, I=I))
self.assertLessEqual(np.sum(x - Xh), 1e-10)
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)
def runTest(self):
m = self.mesh
e = ElementTriP1()
basis = InteriorBasis(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 runTest(self):
m = MeshTri()
e = ElementTriP1()
basis = InteriorBasis(m, e)
@Functional
def feqx(w):
from skfem.helpers import grad
f = w['func'] # f(x) = x
return grad(f)[0] # f'(x) = 1
func = basis.interpolate(m.p[0])
# integrate f'(x) = 1 over [0, 1]^2
self.assertAlmostEqual(feqx.assemble(basis, func=func), 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 = MeshTri()
e = ElementTriP1()
basis = InteriorBasis(m, e)
@Functional
def feqx(w):
from skfem.helpers import grad
f = w['func'] # f(x, y) = x
g = w['gunc'] # g(x, y) = y
return grad(f)[0] + grad(g)[1]
func = basis.interpolate(m.p[0])
gunc = basis.interpolate(m.p[1])
self.assertAlmostEqual(feqx.assemble(basis, func=func, gunc=gunc), 2.)
def runTest(self):
m = MeshTri()
e = ElementTriArgyris()
basis = InteriorBasis(m, e)
all_dofs = basis.get_dofs()
self.assertEqual(len(all_dofs.keep('u').nodal), 1)
self.assertTrue('u' in all_dofs.keep('u').nodal)
self.assertEqual(len(all_dofs.keep('u_n').facet), 1)
self.assertEqual(len(all_dofs.drop('u').facet), 1)
all_dofs = basis.dofs.get_facet_dofs(m.facets_satisfying(lambda x: 1))
self.assertEqual(len(all_dofs.keep('u_n').facet), 1)
self.assertEqual(len(all_dofs.drop('u').facet), 1)
def runTest(self):
m = self.case[0]()
m.refine(self.prerefs)
hs = []
L2s = []
for itr in range(3):
e = self.case[1]()
ib = InteriorBasis(m, e)
@BilinearForm
def bilinf(u, v, w):
return ddot(dd(u), dd(v))
@LinearForm
def linf(v, w):
return 1. * v
K = asm(bilinf, ib)
f = asm(linf, ib)
x = solve(*condense(K, f, D=ib.get_dofs().all()))
X = ib.interpolate(x)
def exact(x):
return (x ** 2 - 2. * x ** 3 + x ** 4) / 24.
@Functional
def error(w):
return (w.w - exact(w.x)) ** 2
L2 = np.sqrt(error.assemble(ib, w=X))
L2s.append(L2)
hs.append(m.param())
m.refine()
hs = np.array(hs)
L2s = np.array(L2s)
pfit = np.polyfit(np.log10(hs), np.log10(L2s), 1)
self.assertGreater(pfit[0], self.limits[0])
self.assertLess(pfit[0], self.limits[1])
self.assertLess(L2s[-1], self.abs_limit)
def runTest(self):
m = MeshLine(np.linspace(0., 1.)).refined(2)
ib = InteriorBasis(m, self.e)
m.define_boundary('left', lambda x: x[0] == 0.0)
m.define_boundary('right', lambda x: x[0] == 1.0)
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)
def test_trace(mtype, e1, e2):
m = mtype().refined(3)
# use the boundary where last coordinate is zero
basis = FacetBasis(
m, e1, facets=m.facets_satisfying(lambda x: x[x.shape[0] - 1] == 0.0))
xfun = projection(lambda x: x[0], InteriorBasis(m, e1))
nbasis, y = basis.trace(xfun,
lambda p: p[0:(p.shape[0] - 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 runTest(self):
m = MeshTri()
e = ElementTriArgyris()
basis = InteriorBasis(m, e)
all_dofs = basis.get_dofs()
assert_allclose(
all_dofs.keep(['u', 'u_n']).keep('u'), all_dofs.keep('u'))
assert_allclose(
all_dofs.drop(['u_x', 'u_y', 'u_xx', 'u_xy', 'u_yy', 'u_n']),
all_dofs.keep('u'))
assert_allclose(all_dofs, all_dofs.drop('does_not_exist'))
assert_allclose(np.empty((0, ), dtype=np.int64),
all_dofs.keep('does_not_exist'))
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)