def get_poisson_steps_scikitfem(points, cells, tol):
from skfem import (
MeshTri,
MeshTet,
InteriorBasis,
ElementTriP1,
ElementTetP1,
asm,
condense,
)
from skfem.models.poisson import laplace, unit_load
import krypy
if cells.shape[1] == 3:
mesh = MeshTri(points.T, cells.T)
e = ElementTriP1()
else:
assert cells.shape[1] == 4
mesh = MeshTet(points.T, cells.T)
e = ElementTetP1()
basis = InteriorBasis(mesh, e)
# assemble
A = asm(laplace, basis)
b = asm(unit_load, basis)
A, b = condense(A, b, I=mesh.interior_nodes(), expand=False)
_, info = krypy.cg(A, b, tol=tol)
return info.iter
def runTest(self):
@LinearForm
def load(v, w):
x = w.x
return (np.sin(np.pi * x[0]) * np.sin(np.pi * x[1]) *
(2.0 * np.pi**2) * v)
m = self.mesh
Nitrs = 9
L2s = np.zeros(Nitrs)
H1s = np.zeros(Nitrs)
for itr in range(Nitrs):
ib = self.create_basis(m, itr + 2)
A = asm(laplace, ib)
b = asm(load, ib)
D = ib.find_dofs()
x = solve(*condense(A, b, D=D))
# calculate error
L2s[itr], H1s[itr] = self.compute_error(m, ib, x)
self.assertLess(H1s[-1], 1e-10)
self.assertLess(L2s[-1], 1e-11)
def s(arr: spmatrix, b: np.ndarray) -> LinearSolver:
ml = ruge_stuben_solver(
skfem.condense(arr, D=dirichlet[v], expand=False))
def f(_: LinearOperator, b: np.ndarray) -> np.ndarray:
return ml.solve(b)
return f
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 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 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 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):
@LinearForm
def load(v, w):
x = w.x
if x.shape[0] == 1:
return (np.sin(np.pi * x[0]) * (np.pi**2) * v)
elif x.shape[0] == 2:
return (np.sin(np.pi * x[0]) * np.sin(np.pi * x[1]) *
(2.0 * np.pi**2) * v)
elif x.shape[0] == 3:
return (np.sin(np.pi * x[0]) * np.sin(np.pi * x[1]) *
np.sin(np.pi * x[2]) * (3.0 * np.pi**2) * v)
else:
raise Exception("The dimension not supported")
m = self.mesh
Nitrs = 3
L2s = np.zeros(Nitrs)
H1s = np.zeros(Nitrs)
hs = np.zeros(Nitrs)
for itr in range(Nitrs):
if itr > 0:
m = self.do_refined(m, itr)
ib = self.create_basis(m)
A = asm(laplace, ib)
b = asm(load, ib)
D = self.get_bc_nodes(ib)
x = solve(*condense(A, b, D=D))
# calculate error
L2s[itr], H1s[itr] = self.compute_error(m, ib, x)
hs[itr] = m.param()
rateL2 = np.polyfit(np.log(hs), np.log(L2s), 1)[0]
rateH1 = np.polyfit(np.log(hs), np.log(H1s), 1)[0]
self.assertLess(np.abs(rateL2 - self.rateL2),
self.eps,
msg='observed L2 rate: {}'.format(rateL2))
self.assertLess(np.abs(rateH1 - self.rateH1),
self.eps,
msg='observed H1 rate: {}'.format(rateH1))
self.assertLess(H1s[-1], 0.3)
self.assertLess(L2s[-1], 0.008)
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.))
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 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_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 runTest(self):
@bilinear_form
def dudv(u, du, v, dv, w):
return sum(du * dv)
m = self.mesh()
m.refine(4)
ib = InteriorBasis(m, self.elem())
A = asm(dudv, 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, 0 * x, x=x, I=I))
self.assertLessEqual(np.sum(x - Xh), 1e-10)
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 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())
gamma = 0.4 * delta**2
beta = 1.25
lambda_ = beta
g = gamma
mesh = MeshTet.init_tensor(np.linspace(0, L, 10 + 1), np.linspace(0, W, 3 + 1),
np.linspace(0, W, 3 + 1))
basis = InteriorBasis(mesh, ElementVectorH1(ElementTetP1()))
clamped = basis.get_dofs(lambda x: x[0] == 0.0).all()
free = basis.complement_dofs(clamped)
K = asm(linear_elasticity(lambda_, mu), basis)
@LinearForm
def load(v, w):
return -rho * g * v.value[2]
f = asm(load, basis)
u = solve(*condense(K, f, I=free))
deformed = MeshTet(mesh.p + u[basis.nodal_dofs], mesh.t)
deformed.save("deformed.xdmf")
u_magnitude = np.linalg.norm(u.reshape((-1, 3)), axis=1)
print("min/max u:", min(u_magnitude), max(u_magnitude))
with TimeSeriesWriter(Path(__file__).with_suffix(".xdmf")) as writer:
writer.write_points_cells(embed(mesh.p.T), {"triangle": mesh.t.T})
t = 0.0
while t < t_final:
t += dt
# Step 1: momentum prediction
uv = skfem.solve(
*skfem.condense(
K_lhs,
K_rhs @ uv_ - P @ (2 * p_ - p__) - skfem.asm(
acceleration, basis["u"], wind=basis["u"].interpolate(u)),
uv0,
D=dirichlet["u"],
),
solver=solvers[0],
)
# Step 2: pressure correction
dp = skfem.solve(*skfem.condense(L["p"], (B / dt) @ uv,
D=dirichlet["p"]),
solver=solvers[1])
# Step 3: velocity correction
p = p_ + dp
def runTest(self):
m = self.case[0]().refined(self.prerefs)
hs = []
L2s = []
for itr in range(3):
e = self.case[1]()
ib = InteriorBasis(m, e)
t = 1.
E = 1.
nu = 0.3
D = E * t ** 3 / (12. * (1. - nu ** 2))
@BilinearForm
def bilinf(u, v, w):
def C(T):
trT = T[0, 0] + T[1, 1]
return E / (1. + nu) * \
np.array([[T[0, 0] + nu / (1. - nu) * trT, T[0, 1]],
[T[1, 0], T[1, 1] + nu / (1. - nu) * trT]])
return t ** 3 / 12.0 * ddot(C(dd(u)), dd(v))
def load(x):
return np.sin(np.pi * x[0]) * np.sin(np.pi * x[1])
@LinearForm
def linf(v, w):
return load(w.x) * v
K = asm(bilinf, ib)
f = asm(linf, ib)
# TODO fix boundary conditions
# u_x should be zero on top/bottom
# u_y should be zero on left/right
x = solve(*condense(K, f, D=ib.get_dofs().all('u')))
X = ib.interpolate(x)
def exact(x):
return 1. / (4. * D * np.pi ** 4) * load(x)
@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 = m.refined()
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)
uv0 = np.zeros(basis["u"].N)
inlet_dofs = basis["u"].get_dofs(mesh.boundaries["inlet"]).all("u^1")
inlet_y_lim = [p.x[1] for p in channel.lines[3].points[::-1]]
monic = np.polynomial.polynomial.Polynomial.fromroots(inlet_y_lim)
uv0[inlet_dofs] = -6 * monic(basis["u"].doflocs[1, inlet_dofs]) / inlet_y_lim[1] ** 2
def embed(xy: np.ndarray) -> np.ndarray:
return np.pad(xy, ((0, 0), (0, 1)), "constant")
solvers = [
pyamgcl.solver(
pyamgcl.amg(
skfem.condense(K_lhs, D=dirichlet["u"], expand=False),
{"coarsening.type": "aggregation", "relax.type": "ilu0"},
),
{"type": "cg"},
),
pyamgcl.solver(
pyamgcl.amg(skfem.condense(L["p"], D=dirichlet["p"], expand=False)),
{"type": "cg"},
),
pyamgcl.solver(
pyamgcl.amg(
skfem.condense(M / dt, D=dirichlet["u"], expand=False),
{"relax.type": "gauss_seidel"},
),
{"type": "cg"},
),
t = 0
with TimeSeriesWriter("channel.xdmf") as writer:
writer.write_points_cells(mesh.p.T, {"triangle": mesh.t.T})
while t < t_final:
t += dt
# Step 1: Momentum prediction (Ern & Guermond 2002, eq. 7.40, p. 274)
uv = skfem.solve(*skfem.condense(
K,
(M / dt) @ uv_ - P @ (2 * p_ - p__),
np.zeros_like(uv_),
D=dirichlet["u"],
))
# Step 2: Projection (Ern & Guermond 2002, eq. 7.41, p. 274)
dp = np.zeros(basis["p"].N)
dp[inlet_pressure_dofs] = p_inlet - p_[inlet_pressure_dofs]
dp = skfem.solve(*skfem.condense(L["p"], B @ uv, dp, D=dirichlet["p"]))
# Step 3: Recover pressure and velocity (E. & G. 2002, p. 274)
p = p_ + dp
print(min(p), "<= p <= ", max(p))