def test_regular_3D(backend, forward_output):
if (backend == 'netcdf4' and forward_output is True) or skip[backend]:
return
K0 = FunctionSpace(N[0], 'F', dtype='D', domain=(0, np.pi))
K1 = FunctionSpace(N[1], 'F', dtype='d', domain=(0, 2 * np.pi))
K2 = FunctionSpace(N[2], 'C', dtype='d', bc=(0, 0))
T = TensorProductSpace(comm, (K0, K1, K2))
filename = 'test3Dr_{}'.format(ex[forward_output])
hfile = writer(filename, T, backend=backend)
if forward_output:
u = Function(T)
else:
u = Array(T)
u[:] = np.random.random(u.shape)
for i in range(3):
hfile.write(i, {
'u':
[u, (u, [slice(None), 4, slice(None)]), (u, [slice(None), 4, 4])]
},
forward_output=forward_output)
if not forward_output and backend == 'hdf5' and comm.Get_rank() == 0:
generate_xdmf(filename + '.h5')
u0 = Function(T) if forward_output else Array(T)
read = reader(filename, T, backend=backend)
read.read(u0, 'u', forward_output=forward_output, step=1)
assert np.allclose(u0, u)
cleanup()
def test_biharmonic2D(family, axis):
la = lla
if family == 'chebyshev':
la = cla
N = (16, 16)
SD = FunctionSpace(N[axis], family=family, bc='Biharmonic')
K1 = FunctionSpace(N[(axis + 1) % 2], family='F', dtype='d')
subcomms = mpi4py_fft.pencil.Subcomm(MPI.COMM_WORLD, allaxes2D[axis])
bases = [K1]
bases.insert(axis, SD)
T = TensorProductSpace(subcomms, bases, axes=allaxes2D[axis])
u = shenfun.TrialFunction(T)
v = shenfun.TestFunction(T)
if family == 'chebyshev':
mat = inner(v, div(grad(div(grad(u)))))
else:
mat = inner(div(grad(v)), div(grad(u)))
H = la.Biharmonic(*mat)
u = Function(T)
u[:] = np.random.random(u.shape) + 1j * np.random.random(u.shape)
f = Function(T)
f = H.matvec(u, f)
g0 = Function(T)
g1 = Function(T)
g2 = Function(T)
M = {d.get_key(): d for d in mat}
g0 = M['SBBmat'].matvec(u, g0)
g1 = M['ABBmat'].matvec(u, g1)
g2 = M['BBBmat'].matvec(u, g2)
assert np.linalg.norm(f - (g0 + g1 + g2)) < 1e-8
def test_solve(quad):
SD = Basis(N, 'C', bc=(0, 0), quad=quad, plan=True)
u = TrialFunction(SD)
v = TestFunction(SD)
A = inner(div(grad(u)), v)
b = np.ones(N)
u_hat = Function(SD)
u_hat = A.solve(b, u=u_hat)
w_hat = Function(SD)
B = SparseMatrix(dict(A), (N - 2, N - 2))
w_hat[:-2] = B.solve(b[:-2], w_hat[:-2])
assert np.all(abs(w_hat[:-2] - u_hat[:-2]) < 1e-8)
ww = w_hat[:-2].repeat(N - 2).reshape((N - 2, N - 2))
bb = b[:-2].repeat(N - 2).reshape((N - 2, N - 2))
ww = B.solve(bb, ww, axis=0)
assert np.all(
abs(ww - u_hat[:-2].repeat(N - 2).reshape((N - 2, N - 2))) < 1e-8)
bb = bb.transpose()
ww = B.solve(bb, ww, axis=1)
assert np.all(
abs(ww - u_hat[:-2].repeat(N - 2).reshape(
(N - 2, N - 2)).transpose()) < 1e-8)
def test_quasiGalerkin(basis):
N = 40
T = FunctionSpace(N, 'C')
S = FunctionSpace(N, 'C', basis=basis)
u = TrialFunction(S)
v = TestFunction(T)
A = inner(v, div(grad(u)))
B = inner(v, u)
Q = chebyshev.quasi.QIGmat(N)
A = Q*A
B = Q*B
M = B-A
sol = la.Solve(M, S)
f_hat = inner(v, Array(T, buffer=fe))
f_hat[:-2] = Q.diags('csc')*f_hat[:-2]
u_hat = Function(S)
u_hat = sol(f_hat, u_hat)
uj = u_hat.backward()
ua = Array(S, buffer=ue)
bb = Q*np.ones(N)
assert abs(np.sum(bb[:8])) < 1e-8
if S.boundary_condition().lower() == 'neumann':
xj, wj = S.points_and_weights()
ua -= np.sum(ua*wj)/np.pi # normalize
uj -= np.sum(uj*wj)/np.pi # normalize
assert np.sqrt(inner(1, (uj-ua)**2)) < 1e-5
def interp(self, y, eigvals, eigvectors, eigval=1, verbose=False):
"""Interpolate solution eigenvector and it's derivative onto y
Parameters
----------
y : array
Interpolation points
eigvals : array
All computed eigenvalues
eigvectors : array
All computed eigenvectors
eigval : int, optional
The chosen eigenvalue, ranked with descending imaginary
part. The largest imaginary part is 1, the second
largest is 2, etc.
verbose : bool, optional
Print information or not
"""
nx, eigval = self.get_eigval(eigval, eigvals, verbose)
SB = Basis(self.N, 'C', bc='Biharmonic', quad=self.quad, dtype='D')
phi_hat = Function(SB)
phi_hat[:-4] = np.squeeze(eigvectors[:, nx])
phi = phi_hat.eval(y)
dphidy = Dx(phi_hat, 0, 1).eval(y)
return eigval, phi, dphidy
def test_curl_cc():
theta, phi = sp.symbols('x,y', real=True, positive=True)
psi = (theta, phi)
r = 1
rv = (r * sp.sin(theta) * sp.cos(phi), r * sp.sin(theta) * sp.sin(phi),
r * sp.cos(theta))
# Manufactured solution
sph = sp.functions.special.spherical_harmonics.Ynm
ue = sph(6, 3, theta, phi)
N, M = 16, 12
L0 = FunctionSpace(N, 'C', domain=(0, np.pi))
F1 = FunctionSpace(M, 'F', dtype='D')
T = TensorProductSpace(comm, (L0, F1), coordinates=(psi, rv))
u_hat = Function(T, buffer=ue)
du = curl(grad(u_hat))
du.terms() == [[]]
r, theta, z = psi = sp.symbols('x,y,z', real=True, positive=True)
rv = (r * sp.cos(theta), r * sp.sin(theta), z)
# Manufactured solution
ue = (r * (1 - r) * sp.cos(4 * theta) - 1 * (r - 1)) * sp.cos(4 * z)
N = 12
F0 = FunctionSpace(N, 'F', dtype='D')
F1 = FunctionSpace(N, 'F', dtype='d')
L = FunctionSpace(N, 'L', bc='Dirichlet', domain=(0, 1))
T = TensorProductSpace(comm, (L, F0, F1), coordinates=(psi, rv))
T1 = T.get_orthogonal()
V = VectorSpace(T1)
u_hat = Function(T, buffer=ue)
du = project(curl(grad(u_hat)), V)
assert np.linalg.norm(du) < 1e-10
def test_project_2dirichlet(quad):
x, y = symbols("x,y")
ue = (cos(4*y)*sin(2*x))*(1-x**2)*(1-y**2)
sizes = (18, 17)
D0 = lbases.ShenDirichlet(sizes[0], quad=quad)
D1 = lbases.ShenDirichlet(sizes[1], quad=quad)
B0 = lbases.Orthogonal(sizes[0], quad=quad)
B1 = lbases.Orthogonal(sizes[1], quad=quad)
DD = TensorProductSpace(comm, (D0, D1))
BD = TensorProductSpace(comm, (B0, D1))
DB = TensorProductSpace(comm, (D0, B1))
BB = TensorProductSpace(comm, (B0, B1))
X = DD.local_mesh(True)
uh = Function(DD, buffer=ue)
dudx_hat = project(Dx(uh, 0, 1), BD)
dx = Function(BD, buffer=ue.diff(x, 1))
assert np.allclose(dx, dudx_hat, 0, 1e-5)
dudy = project(Dx(uh, 1, 1), DB).backward()
duedy = Array(DB, buffer=ue.diff(y, 1))
assert np.allclose(duedy, dudy, 0, 1e-5)
us_hat = Function(BB)
uq = uh.backward()
us = project(uq, BB, output_array=us_hat).backward()
assert np.allclose(us, uq, 0, 1e-5)
dudxy = project(Dx(us_hat, 0, 1) + Dx(us_hat, 1, 1), BB).backward()
dxy = Array(BB, buffer=ue.diff(x, 1) + ue.diff(y, 1))
assert np.allclose(dxy, dudxy, 0, 1e-5), np.linalg.norm(dxy-dudxy)
def divergenceConvection(rhs, u_dealias, u_hat, K, VFSp, FSTp, FSBp, FCTp, work,
mat, la, add=False):
"""c_i = div(u_i u_j)"""
if not add:
rhs.fill(0)
F_tmp = Function(VFSp, buffer=work[(rhs, 0, True)])
F_tmp2 = Function(VFSp, buffer=work[(rhs, 1, True)])
U = u_dealias
F_tmp[0] = FSTp.forward(U[0]*U[0], F_tmp[0])
F_tmp[1] = FSTp.forward(U[0]*U[1], F_tmp[1])
F_tmp[2] = FSTp.forward(U[0]*U[2], F_tmp[2])
F_tmp2 = project(Dx(F_tmp, 0, 1), VFSp, output_array=F_tmp2)
rhs += F_tmp2
F_tmp2[0] = FSTp.forward(U[0]*U[1], F_tmp2[0])
F_tmp2[1] = FSTp.forward(U[0]*U[2], F_tmp2[1])
rhs[0] += 1j*K[1]*F_tmp2[0] # duvdy
rhs[0] += 1j*K[2]*F_tmp2[1] # duwdz
F_tmp[0] = FSTp.forward(U[1]*U[1], F_tmp[0])
F_tmp[1] = FSTp.forward(U[1]*U[2], F_tmp[1])
F_tmp[2] = FSTp.forward(U[2]*U[2], F_tmp[2])
rhs[1] += 1j*K[1]*F_tmp[0] # dvvdy
rhs[1] += 1j*K[2]*F_tmp[1] # dvwdz
rhs[2] += 1j*K[1]*F_tmp[1] # dvwdy
rhs[2] += 1j*K[2]*F_tmp[2] # dwwdz
return rhs
def test_mixed_2D(backend, forward_output, as_scalar):
if (backend == 'netcdf4' and forward_output is True) or skip[backend]:
return
K0 = FunctionSpace(N[0], 'F')
K1 = FunctionSpace(N[1], 'C')
T = TensorProductSpace(comm, (K0, K1))
TT = CompositeSpace([T, T])
filename = 'test2Dm_{}'.format(ex[forward_output])
hfile = writer(filename, TT, backend=backend)
if forward_output:
uf = Function(TT, val=2)
else:
uf = Array(TT, val=2)
hfile.write(0, {'uf': [uf]}, as_scalar=as_scalar)
hfile.write(1, {'uf': [uf]}, as_scalar=as_scalar)
if not forward_output and backend == 'hdf5' and comm.Get_rank() == 0:
generate_xdmf(filename + '.h5')
if as_scalar is False:
u0 = Function(TT) if forward_output else Array(TT)
read = reader(filename, TT, backend=backend)
read.read(u0, 'uf', step=1)
assert np.allclose(u0, uf)
else:
u0 = Function(T) if forward_output else Array(T)
read = reader(filename, T, backend=backend)
read.read(u0, 'uf0', step=1)
assert np.allclose(u0, uf[0])
cleanup()
def test_helmholtz2D(family, axis):
la = lla
if family == 'chebyshev':
la = cla
N = (8, 9)
SD = FunctionSpace(N[axis], family=family, bc=(0, 0))
K1 = FunctionSpace(N[(axis+1)%2], family='F', dtype='d')
subcomms = mpi4py_fft.pencil.Subcomm(MPI.COMM_WORLD, allaxes2D[axis])
bases = [K1]
bases.insert(axis, SD)
T = TensorProductSpace(subcomms, bases, axes=allaxes2D[axis], modify_spaces_inplace=True)
u = shenfun.TrialFunction(T)
v = shenfun.TestFunction(T)
if family == 'chebyshev':
mat = inner(v, div(grad(u)))
else:
mat = inner(grad(v), grad(u))
H = la.Helmholtz(*mat)
u = Function(T)
s = SD.sl[SD.slice()]
u[s] = np.random.random(u[s].shape) + 1j*np.random.random(u[s].shape)
f = Function(T)
f = H.matvec(u, f)
g0 = Function(T)
g1 = Function(T)
M = {d.get_key(): d for d in mat}
g0 = M['ADDmat'].matvec(u, g0)
g1 = M['BDDmat'].matvec(u, g1)
assert np.linalg.norm(f-(g0+g1)) < 1e-12, np.linalg.norm(f-(g0+g1))
uc = Function(T)
uc = H(uc, f)
assert np.linalg.norm(uc-u) < 1e-12
def test_mixed_3D(backend, forward_output, as_scalar):
if (backend == 'netcdf4' and forward_output is True) or skip[backend]:
return
K0 = FunctionSpace(N[0], 'F', dtype='D', domain=(0, np.pi))
K1 = FunctionSpace(N[1], 'F', dtype='d', domain=(0, 2 * np.pi))
K2 = FunctionSpace(N[2], 'C')
T = TensorProductSpace(comm, (K0, K1, K2))
TT = VectorSpace(T)
filename = 'test3Dm_{}'.format(ex[forward_output])
hfile = writer(filename, TT, backend=backend)
uf = Function(TT, val=2) if forward_output else Array(TT, val=2)
uf[0] = 1
data = {
'ux': (uf[0], (uf[0], [slice(None), 4,
slice(None)]), (uf[0], [slice(None), 4, 4])),
'uy': (uf[1], (uf[1], [slice(None), 4,
slice(None)]), (uf[1], [slice(None), 4, 4])),
'u': [uf, (uf, [slice(None), 4, slice(None)])]
}
hfile.write(0, data, as_scalar=as_scalar)
hfile.write(1, data, as_scalar=as_scalar)
if not forward_output and backend == 'hdf5' and comm.Get_rank() == 0:
generate_xdmf(filename + '.h5')
if as_scalar is False:
u0 = Function(TT) if forward_output else Array(TT)
read = reader(filename, TT, backend=backend)
read.read(u0, 'u', step=1)
assert np.allclose(u0, uf)
else:
u0 = Function(T) if forward_output else Array(T)
read = reader(filename, T, backend=backend)
read.read(u0, 'u0', step=1)
assert np.allclose(u0, uf[0])
cleanup()
def __init__(self, T, L=None, N=None, update=None, **params):
IntegratorBase.__init__(self, T, L=L, N=N, update=update, **params)
self.U_hat0 = Function(T)
self.U_hat1 = Function(T)
self.dU = Function(T)
self.a = np.array([1. / 6., 1. / 3., 1. / 3., 1. / 6.])
self.b = np.array([0.5, 0.5, 1.])
def test_backward_uniform(family):
T = FunctionSpace(N, family)
uT = Function(T, buffer=f)
ub = uT.backward(kind='uniform')
xj = T.mesh(uniform=True)
fj = sp.lambdify(x, f)(xj)
assert np.linalg.norm(fj - ub) < 1e-8
def __init__(self, T, L=None, N=None, update=None, **params):
IntegratorBase.__init__(self, T, L=L, N=N, update=update, **params)
self.dU = Function(T)
self.dU1 = Function(T)
self.a = (8. / 15., 5. / 12., 3. / 4.)
self.b = (0.0, -17. / 60., -5. / 12.)
self.c = (0., 8. / 15., 2. / 3., 1)
self.solver = None
self.rhs_mats = None
self.w0 = Function(self.T).v
self.mask = self.T.get_mask_nyquist()
def __init__(self, T, L=None, N=None, update=None, **params):
IntegratorBase.__init__(self, T, L=L, N=N, update=update, **params)
self.U_hat0 = Function(T)
self.U_hat1 = Function(T)
self.dU = Function(T)
self.dU0 = Function(T)
self.V2 = Function(T)
self.psi = np.zeros((4, ) + self.U_hat0.shape, dtype=np.float)
self.a = None
self.b = [0.5, 0.5, 0.5]
self.ehL = None
self.ehL_h = None
def test_backward2ND():
T0 = FunctionSpace(N, 'C')
L0 = FunctionSpace(N, 'L')
T1 = FunctionSpace(N, 'C')
L1 = FunctionSpace(N, 'L')
TT = TensorProductSpace(comm, (T0, T1))
LL = TensorProductSpace(comm, (L0, L1))
uT = Function(TT, buffer=h)
uL = Function(LL, buffer=h)
u2 = uL.backward(kind=TT)
uT2 = project(u2, TT)
assert np.linalg.norm(uT2 - uT)
def test_curl2():
# Test projection of curl
K0 = FunctionSpace(N[0], 'C', bc=(0, 0))
K1 = FunctionSpace(N[1], 'F', dtype='D')
K2 = FunctionSpace(N[2], 'F', dtype='d')
K3 = FunctionSpace(N[0], 'C')
T = TensorProductSpace(comm, (K0, K1, K2))
TT = TensorProductSpace(comm, (K3, K1, K2))
X = T.local_mesh(True)
K = T.local_wavenumbers(False)
Tk = VectorSpace(T)
TTk = VectorSpace([T, T, TT])
U = Array(Tk)
U_hat = Function(Tk)
curl_hat = Function(TTk)
curl_ = Array(TTk)
# Initialize a Taylor Green vortex
U[0] = np.sin(X[0]) * np.cos(X[1]) * np.cos(X[2]) * (1 - X[0]**2)
U[1] = -np.cos(X[0]) * np.sin(X[1]) * np.cos(X[2]) * (1 - X[0]**2)
U[2] = 0
U_hat = Tk.forward(U, U_hat)
Uc = U_hat.copy()
U = Tk.backward(U_hat, U)
U_hat = Tk.forward(U, U_hat)
assert allclose(U_hat, Uc)
# Compute curl first by computing each term individually
curl_hat[0] = 1j * (K[1] * U_hat[2] - K[2] * U_hat[1])
curl_[0] = T.backward(
curl_hat[0], curl_[0]) # No x-derivatives, still in Dirichlet space
dwdx_hat = project(Dx(U_hat[2], 0, 1), TT) # Need to use space without bc
dvdx_hat = project(Dx(U_hat[1], 0, 1), TT) # Need to use space without bc
dwdx = Array(TT)
dvdx = Array(TT)
dwdx = TT.backward(dwdx_hat, dwdx)
dvdx = TT.backward(dvdx_hat, dvdx)
curl_hat[1] = 1j * K[2] * U_hat[0]
curl_hat[2] = -1j * K[1] * U_hat[0]
curl_[1] = T.backward(curl_hat[1], curl_[1])
curl_[2] = T.backward(curl_hat[2], curl_[2])
curl_[1] -= dwdx
curl_[2] += dvdx
# Now do it with project
w_hat = project(curl(U_hat), TTk)
w = Array(TTk)
w = TTk.backward(w_hat, w)
assert allclose(w, curl_)
def test_backward3D():
T = FunctionSpace(N, 'C')
L = FunctionSpace(N, 'L')
F0 = FunctionSpace(N, 'F', dtype='D')
F1 = FunctionSpace(N, 'F', dtype='d')
TT = TensorProductSpace(comm, (F0, T, F1))
TL = TensorProductSpace(comm, (F0, L, F1))
uT = Function(TT, buffer=h)
uL = Function(TL, buffer=h)
u2 = uL.backward(kind=TT)
uT2 = project(u2, TT)
assert np.linalg.norm(uT2 - uT)
def test_project_2dirichlet(quad):
x, y = symbols("x,y")
ue = (cos(4 * y) * sin(2 * x)) * (1 - x**2) * (1 - y**2)
sizes = (25, 24)
D0 = lbases.ShenDirichletBasis(sizes[0], quad=quad)
D1 = lbases.ShenDirichletBasis(sizes[1], quad=quad)
B0 = lbases.Basis(sizes[0], quad=quad)
B1 = lbases.Basis(sizes[1], quad=quad)
DD = TensorProductSpace(comm, (D0, D1))
BD = TensorProductSpace(comm, (B0, D1))
DB = TensorProductSpace(comm, (D0, B1))
BB = TensorProductSpace(comm, (B0, B1))
X = DD.local_mesh(True)
ul = lambdify((x, y), ue, 'numpy')
uq = Array(DD)
uq[:] = ul(*X)
uh = Function(DD)
uh = DD.forward(uq, uh)
dudx_hat = project(Dx(uh, 0, 1), BD)
dudx = Array(BD)
dudx = BD.backward(dudx_hat, dudx)
duedx = ue.diff(x, 1)
duxl = lambdify((x, y), duedx, 'numpy')
dx = duxl(*X)
assert np.allclose(dx, dudx)
dudy_hat = project(Dx(uh, 1, 1), DB)
dudy = Array(DB)
dudy = DB.backward(dudy_hat, dudy)
duedy = ue.diff(y, 1)
duyl = lambdify((x, y), duedy, 'numpy')
dy = duyl(*X)
assert np.allclose(dy, dudy)
us_hat = Function(BB)
us_hat = project(uq, BB, output_array=us_hat)
us = Array(BB)
us = BB.backward(us_hat, us)
assert np.allclose(us, uq)
dudxy_hat = project(Dx(us_hat, 0, 1) + Dx(us_hat, 1, 1), BB)
dudxy = Array(BB)
dudxy = BB.backward(dudxy_hat, dudxy)
duedxy = ue.diff(x, 1) + ue.diff(y, 1)
duxyl = lambdify((x, y), duedxy, 'numpy')
dxy = duxyl(*X)
assert np.allclose(dxy, dudxy)
def test_refine():
assert comm.Get_size() < 7
N = (8, 9, 10)
F0 = Basis(8, 'F', dtype='D')
F1 = Basis(9, 'F', dtype='D')
F2 = Basis(10, 'F', dtype='d')
T = TensorProductSpace(comm, (F0, F1, F2), slab=True, collapse_fourier=True)
u_hat = Function(T)
u = Array(T)
u[:] = np.random.random(u.shape)
u_hat = u.forward(u_hat)
Tp = T.get_dealiased(padding_factor=(2, 2, 2))
u_ = Array(Tp)
up_hat = Function(Tp)
assert up_hat.commsizes == u_hat.commsizes
u2 = u_hat.refine(2*np.array(N))
V = VectorTensorProductSpace(T)
u_hat = Function(V)
u = Array(V)
u[:] = np.random.random(u.shape)
u_hat = u.forward(u_hat)
Vp = V.get_dealiased(padding_factor=(2, 2, 2))
u_ = Array(Vp)
up_hat = Function(Vp)
assert up_hat.commsizes == u_hat.commsizes
u3 = u_hat.refine(2*np.array(N))
def test_eval_expression():
import sympy as sp
from shenfun import div, grad
x, y, z = sp.symbols('x,y,z')
B0 = FunctionSpace(16, 'C')
B1 = FunctionSpace(17, 'C')
B2 = FunctionSpace(20, 'F', dtype='d')
TB = TensorProductSpace(comm, (B0, B1, B2))
f = sp.sin(x)+sp.sin(y)+sp.sin(z)
dfx = f.diff(x, 2) + f.diff(y, 2) + f.diff(z, 2)
fa = Function(TB, buffer=f)
dfe = div(grad(fa))
dfa = project(dfe, TB)
xyz = np.array([[0.25, 0.5, 0.75],
[0.25, 0.5, 0.75],
[0.25, 0.5, 0.75]])
f0 = lambdify((x, y, z), dfx)(*xyz)
f1 = dfe.eval(xyz)
f2 = dfa.eval(xyz)
assert np.allclose(f0, f1, 1e-7)
assert np.allclose(f1, f2, 1e-7)
def test_energy_fourier(N):
B0 = FunctionSpace(N[0], 'F', dtype='D')
B1 = FunctionSpace(N[1], 'F', dtype='D')
B2 = FunctionSpace(N[2], 'F', dtype='d')
for bases, axes in zip(((B0, B1, B2), (B0, B2, B1)),
((0, 1, 2), (2, 0, 1))):
T = TensorProductSpace(comm, bases, axes=axes)
u_hat = Function(T)
u_hat[:] = np.random.random(
u_hat.shape) + 1j * np.random.random(u_hat.shape)
u = u_hat.backward()
u_hat = u.forward(u_hat)
u = u_hat.backward(u)
e0 = comm.allreduce(np.sum(u.v * u.v) / np.prod(N))
e1 = fourier.energy_fourier(u_hat, T)
assert abs(e0 - e1) < 1e-10
def get_adaptive(self, fun=None, reltol=1e-12, abstol=1e-15):
"""Return space (otherwise as self) with number of quadrature points
determined by fitting `fun`
Returns
-------
SpectralBase
A new space with adaptively found number of quadrature points
"""
from shenfun import Function
assert isinstance(fun, sp.Expr)
assert self.N == 0
T = self.get_refined(5)
converged = False
count = 0
points = np.random.random(8)
points = T.domain[0] + points * (T.domain[1] - T.domain[0])
sym = fun.free_symbols
assert len(sym) == 1
x = sym.pop()
fx = sp.lambdify(x, fun)
while (not converged) and count < 12:
T = T.get_refined(int(1.7 * T.N))
u = Function(T, buffer=fun)
res = T.eval(points, u)
exact = fx(points)
energy = np.linalg.norm(res - exact)
#print(T.N, energy)
converged = energy**2 < abstol
count += 1
# trim trailing zeros (if any)
trailing_zeros = T.count_trailing_zeros(u, reltol, abstol)
T = T.get_refined(T.N - trailing_zeros)
return T
def test_project(typecode, dim, ST, quad):
# Using sympy to compute an analytical solution
x, y, z = symbols("x,y,z")
sizes = (25, 24)
funcs = {
(1, 0): (cos(4 * y) * sin(2 * np.pi * x)) * (1 - x**2),
(1, 1): (cos(4 * x) * sin(2 * np.pi * y)) * (1 - y**2),
(2, 0): (sin(6 * z) * cos(4 * y) * sin(2 * np.pi * x)) * (1 - x**2),
(2, 1): (sin(2 * z) * cos(4 * x) * sin(2 * np.pi * y)) * (1 - y**2),
(2, 2): (sin(2 * x) * cos(4 * y) * sin(2 * np.pi * z)) * (1 - z**2)
}
syms = {1: (x, y), 2: (x, y, z)}
xs = {0: x, 1: y, 2: z}
for shape in product(*([sizes] * dim)):
bases = []
for n in shape[:-1]:
bases.append(Basis(n, 'F', dtype=typecode.upper()))
bases.append(Basis(shape[-1], 'F', dtype=typecode))
if dim < 3:
n = min(shape)
if typecode in 'fdg':
n //= 2
n += 1
if n < comm.size:
continue
for axis in range(dim + 1):
ST0 = ST(shape[-1], quad=quad)
bases.insert(axis, ST0)
fft = TensorProductSpace(comm,
bases,
dtype=typecode,
axes=axes[dim][axis])
X = fft.local_mesh(True)
ue = funcs[(dim, axis)]
ul = lambdify(syms[dim], ue, 'numpy')
uq = ul(*X).astype(typecode)
uh = Function(fft)
uh = fft.forward(uq, uh)
due = ue.diff(xs[axis], 1)
dul = lambdify(syms[dim], due, 'numpy')
duq = dul(*X).astype(typecode)
uf = project(Dx(uh, axis, 1), fft)
uy = Array(fft)
uy = fft.backward(uf, uy)
assert np.allclose(uy, duq, 0, 1e-6)
for ax in (x for x in range(dim + 1) if x is not axis):
due = ue.diff(xs[axis], 1, xs[ax], 1)
dul = lambdify(syms[dim], due, 'numpy')
duq = dul(*X).astype(typecode)
uf = project(Dx(Dx(uh, axis, 1), ax, 1), fft)
uy = Array(fft)
uy = fft.backward(uf, uy)
assert np.allclose(uy, duq, 0, 1e-6)
bases.pop(axis)
fft.destroy()
def test_padding_biharmonic(family):
N = 8
B = FunctionSpace(N, family, bc=(0, 0, 0, 0))
Bp = B.get_dealiased(1.5)
u = Function(B)
u[:(N - 4)] = np.random.random(N - 4)
up = Array(Bp)
up = Bp.backward(u, fast_transform=False)
uf = Bp.forward(up, fast_transform=False)
assert np.linalg.norm(uf - u) < 1e-12
if family == 'C':
up = Bp.backward(u)
uf = Bp.forward(up)
assert np.linalg.norm(uf - u) < 1e-12
# Test padding 2D
F = FunctionSpace(N, 'F', dtype='d')
T = TensorProductSpace(comm, (B, F))
Tp = T.get_dealiased(1.5)
u = Function(T)
u[:-4, :-1] = np.random.random(u[:-4, :-1].shape)
up = Tp.backward(u)
uc = Tp.forward(up)
assert np.linalg.norm(u - uc) < 1e-8
# Test padding 3D
F1 = FunctionSpace(N, 'F', dtype='D')
T = TensorProductSpace(comm, (F1, F, B))
Tp = T.get_dealiased(1.5)
u = Function(T)
u[:, :, :-4] = np.random.random(u[:, :, :-4].shape)
u = u.backward().forward() # Clean
up = Tp.backward(u)
uc = Tp.forward(up)
assert np.linalg.norm(u - uc) < 1e-8
def test_padding_neumann(family):
N = 8
B = FunctionSpace(N, family, bc={'left': ('N', 0), 'right': ('N', 0)})
Bp = B.get_dealiased(1.5)
u = Function(B)
u[1:-2] = np.random.random(N - 3)
up = Array(Bp)
up = Bp.backward(u, fast_transform=False)
uf = Bp.forward(up, fast_transform=False)
assert np.linalg.norm(uf - u) < 1e-12
if family == 'C':
up = Bp.backward(u)
uf = Bp.forward(up)
assert np.linalg.norm(uf - u) < 1e-12
# Test padding 2D
F = FunctionSpace(N, 'F', dtype='d')
T = TensorProductSpace(comm, (B, F))
Tp = T.get_dealiased(1.5)
u = Function(T)
u[1:-2, :-1] = np.random.random(u[1:-2, :-1].shape)
up = Tp.backward(u)
uc = Tp.forward(up)
assert np.linalg.norm(u - uc) < 1e-8
# Test padding 3D
F1 = FunctionSpace(N, 'F', dtype='D')
T = TensorProductSpace(comm, (F1, F, B))
Tp = T.get_dealiased(1.5)
u = Function(T)
u[:, :, 1:-2] = np.random.random(u[:, :, 1:-2].shape)
u = u.backward().forward() # Clean
up = Tp.backward(u)
uc = Tp.forward(up)
assert np.linalg.norm(u - uc) < 1e-8
def __init__(self, T,
L=None,
N=None,
update=None,
**params):
IntegratorBase.__init__(self, T, L=L, N=N, update=update, **params)
self.dU = Function(T)
self.psi = None
self.ehL = None
def test_PDMA(quad):
SB = FunctionSpace(N, 'C', bc=(0, 0, 0, 0), quad=quad)
u = TrialFunction(SB)
v = TestFunction(SB)
points, weights = SB.points_and_weights(N)
fj = Array(SB, buffer=np.random.randn(N))
f_hat = Function(SB)
f_hat = inner(v, fj, output_array=f_hat)
A = inner(v, div(grad(u)))
B = inner(v, u)
s = SB.slice()
H = A + B
P = PDMA(A, B, A.scale, B.scale, solver='cython')
u_hat = Function(SB)
u_hat[s] = solve(H.diags().toarray()[s, s], f_hat[s])
u_hat2 = Function(SB)
u_hat2 = P(u_hat2, f_hat)
assert np.allclose(u_hat2, u_hat)
def test_TwoDMA():
N = 12
SD = FunctionSpace(N, 'C', basis='ShenDirichlet')
HH = FunctionSpace(N, 'C', basis='Heinrichs')
u = TrialFunction(HH)
v = TestFunction(SD)
points, weights = SD.points_and_weights(N)
fj = Array(SD, buffer=np.random.randn(N))
f_hat = Function(SD)
f_hat = inner(v, fj, output_array=f_hat)
A = inner(v, div(grad(u)))
sol = TwoDMA(A)
u_hat = Function(HH)
u_hat = sol(f_hat, u_hat)
sol2 = la.Solve(A, HH)
u_hat2 = Function(HH)
u_hat2 = sol2(f_hat, u_hat2)
assert np.allclose(u_hat2, u_hat)
def main(N, family, bci, bcj, plotting=False):
global fe, ue
BX = FunctionSpace(N, family=family, bc=bcx[bci], domain=xdomain)
BY = FunctionSpace(N, family=family, bc=bcy[bcj], domain=ydomain)
T = TensorProductSpace(comm, (BX, BY))
u = TrialFunction(T)
v = TestFunction(T)
# Get f on quad points
fj = Array(T, buffer=fe)
# Compare with analytical solution
ua = Array(T, buffer=ue)
if T.use_fixed_gauge:
mean = dx(ua, weighted=True) / inner(1, Array(T, val=1))
# Compute right hand side of Poisson equation
f_hat = Function(T)
f_hat = inner(v, fj, output_array=f_hat)
# Get left hand side of Poisson equation
A = inner(v, -div(grad(u)))
u_hat = Function(T)
sol = la.Solver2D(A, fixed_gauge=mean if T.use_fixed_gauge else None)
u_hat = sol(f_hat, u_hat)
uj = u_hat.backward()
assert np.allclose(uj, ua), np.linalg.norm(uj - ua)
print("Error=%2.16e" % (np.sqrt(dx((uj - ua)**2))))
if 'pytest' not in os.environ and plotting is True:
import matplotlib.pyplot as plt
X, Y = T.local_mesh(True)
plt.contourf(X, Y, uj, 100)
plt.colorbar()
plt.figure()
plt.contourf(X, Y, ua, 100)
plt.colorbar()
plt.figure()
plt.contourf(X, Y, ua - uj, 100)
plt.colorbar()
