def test_eval_projection(self):
"""
Test validity of the local projection-based kernel.
Choose
u, v in V
k(x,y) in VxV (symmetric/non-symm)
cell01, cell02
Compare
v^T*Kloc*u ?= Icell01 Icell02 k(x,y)u(y)dy dx
"""
#
# 1D
#
# Mesh
mesh = Mesh1D(resolution=(3, ))
# Finite element space
etype = 'Q1'
element = QuadFE(1, etype)
# Dofhandler
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Basis functions
phi = Basis(dofhandler, 'u')
# Symmetric kernel function
kfns = {
'symmetric': lambda x, y: x * y,
'non_symmetric': lambda x, y: x - y
}
vals = {
'symmetric':
(1 / 2 * 1 / 9 - 1 / 3 * 1 / 27) * (1 / 3 - 1 / 3 * 8 / 27),
'non_symmetric': (1 / 18 - 1 / 3 * 1 / 27) * (1 / 2 - 2 / 9) +
(1 / 18 - 1 / 3) * (1 / 3 - 1 / 3 * 8 / 27)
}
for ktype in ['symmetric', 'non_symmetric']:
# Get kernel function
kfn = kfns[ktype]
# Define integral kernel
kernel = Kernel(Explicit(kfn, dim=1, n_variables=2))
# Define Bilinear Form
form = IPForm(kernel, trial=phi, test=phi)
#
# Compute inputs required for evaluating form_loc
#
# Assembler
assembler = Assembler(form, mesh)
# Cells
ci = mesh.cells.get_child(0)
cj = mesh.cells.get_child(2)
# Shape function info on cells
ci_sinfo = assembler.shape_info(ci)
cj_sinfo = assembler.shape_info(cj)
# Gauss nodes and weights on cell
xi_g, wi_g, phii, dofsi = assembler.shape_eval(ci)
xj_g, wj_g, phij, dofsj = assembler.shape_eval(cj)
#
# Evaluate form
#
form_loc = form.eval((ci,cj), (xi_g,xj_g), \
(wi_g,wj_g), (phii,phij), (dofsi,dofsj))
#
# Define functions
#
u = Nodal(f=lambda x: x, basis=phi)
v = Nodal(f=lambda x: 1 - x, basis=phi)
#
# Get local node values
#
# Degrees of freedom
ci_dofs = phi.dofs(ci)
cj_dofs = phi.dofs(cj)
uj = u.data()[np.array(cj_dofs)]
vi = v.data()[np.array(ci_dofs)]
if ktype == 'symmetric':
# Local form by hand
c10 = 1 / 54
c11 = 1 / 27
c20 = 3 / 2 - 1 - 3 / 2 * 4 / 9 + 8 / 27
c21 = 4 / 27
fl = np.array([[c10 * c20, c10 * c21], [c11 * c20, c11 * c21]])
# Compare computed and explicit local forms
self.assertTrue(np.allclose(fl, form_loc))
# Evaluate Ici Icj k(x,y) y dy (1-x)dx
fa = np.dot(vi.T, form_loc.dot(uj))
fe = vals[ktype]
self.assertAlmostEqual(fa, fe)
def test_eval_interpolation(self):
#
# 1D
#
# Mesh
mesh = Mesh1D(resolution=(3, ))
# Finite element space
etype = 'Q1'
element = QuadFE(1, etype)
# Dofhandler
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Basis functions
phi = Basis(dofhandler, 'u')
# Symmetric kernel function
kfns = {
'symmetric': lambda x, y: x * y,
'non_symmetric': lambda x, y: x - y
}
vals = {
'symmetric':
np.array([0, 1 / 9 * (1 - 8 / 27)]),
'non_symmetric':
np.array([1 / 3 * (-1 + 8 / 27), 1 / 6 * 5 / 9 - 1 / 3 * 19 / 27])
}
for ktype in ['symmetric', 'non_symmetric']:
# Get kernel function
kfn = kfns[ktype]
# Define integral kernel
kernel = Kernel(Explicit(kfn, dim=1, n_variables=2))
# Define Bilinear Form
form = IIForm(kernel, trial=phi, test=phi)
#
# Compute inputs required for evaluating form_loc
#
# Assembler
assembler = Assembler(form, mesh)
# Cells
cj = mesh.cells.get_child(2)
ci = mesh.cells.get_child(0)
# Gauss nodes and weights on cell
xj_g, wj_g, phij, dofsj = assembler.shape_eval(cj)
#
# Evaluate form
#
form_loc = form.eval(cj, xj_g, wj_g, phij, dofsj)
#
# Define functions
#
u = Nodal(lambda x: x, basis=phi)
#
# Get local node values
#
# Degrees of freedom
cj_dofs = phi.dofs(cj)
ci_dofs = phi.dofs(ci)
uj = u.data()[np.array(cj_dofs)]
# Evaluate Ici Icj k(x,y) y dy (1-x)dx
fa = form_loc[ci_dofs].dot(uj)
fe = vals[ktype][:, None]
self.assertTrue(np.allclose(fa, fe))