def faceDivy(self):
if (self.dim < 2):
return None
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fy', 'V')
V = self.vol
return sdiag(1 / V) * self._faceDivStencily * sdiag(S)
def faceDivx(self):
"""
Construct divergence operator in the x component (face-stg to
cell-centres).
"""
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fx', 'V')
V = self.vol
return sdiag(1 / V) * self._faceDivStencilx * sdiag(S)
def faceDivz(self):
"""
Construct divergence operator in the z component (face-stg to
cell-centers).
"""
if (self.dim < 3):
return None
# Compute areas of cell faces & volumes
S = self.r(self.area, 'F', 'Fz', 'V')
V = self.vol
return sdiag(1 / V) * self._faceDivStencilz * sdiag(S)
def faceDiv(self):
"""
Construct divergence operator (face-stg to cell-centres).
"""
if getattr(self, '_faceDiv', None) is None:
# Get the stencil of +1, -1's
D = self._faceDivStencil
# Compute areas of cell faces & volumes
S = self.area
V = self.vol
self._faceDiv = sdiag(1 / V) * D * sdiag(S)
return self._faceDiv
def test_zero(self):
z = Zero()
assert z == 0
assert not (z < 0)
assert z <= 0
assert not (z > 0)
assert z >= 0
assert +z == z
assert -z == z
assert z + 1 == 1
assert z + 3 + z == 3
assert z - 3 == -3
assert z - 3 - z == -3
assert 3 * z == 0
assert z * 3 == 0
assert z / 3 == 0
a = 1
a += z
assert a == 1
a = 1
a += z
assert a == 1
self.assertRaises(ZeroDivisionError, lambda: 3 / z)
assert mkvc(z) == 0
assert sdiag(z) * a == 0
assert z.T == 0
assert z.transpose() == 0
def cellGradBC(self):
"""
The cell centered Gradient boundary condition matrix
"""
if getattr(self, '_cellGradBC', None) is None:
BC = self.setCellGradBC(self._cellGradBC_list)
n = self.vnC
if self.dim == 1:
G = ddxCellGradBC(n[0], BC[0])
elif self.dim == 2:
G1 = sp.kron(speye(n[1]), ddxCellGradBC(n[0], BC[0]))
G2 = sp.kron(ddxCellGradBC(n[1], BC[1]), speye(n[0]))
G = sp.block_diag((G1, G2), format="csr")
elif self.dim == 3:
G1 = kron3(speye(n[2]), speye(n[1]),
ddxCellGradBC(n[0], BC[0]))
G2 = kron3(speye(n[2]), ddxCellGradBC(n[1], BC[1]),
speye(n[0]))
G3 = kron3(ddxCellGradBC(n[2], BC[2]), speye(n[1]),
speye(n[0]))
G = sp.block_diag((G1, G2, G3), format="csr")
# Compute areas of cell faces & volumes
S = self.area
V = self.aveCC2F * self.vol # Average volume between adjacent cells
self._cellGradBC = sdiag(S / V) * G
return self._cellGradBC
def _nodalLaplacianx(self):
Hx = sdiag(1. / self.hx)
if self.dim == 2:
Hx = sp.kron(speye(self.nNy), Hx)
elif self.dim == 3:
Hx = kron3(speye(self.nNz), speye(self.nNy), Hx)
return Hx.T * self._nodalGradStencilx * Hx
def getError(self):
# Test function
# PDE: Curl(Curl Ez) + Ez = q
# faces_z are cell_centers on 2D mesh
ez_fun = lambda x: np.cos(np.pi * x[:, 0]) * np.cos(np.pi * x[:, 1])
q_fun = lambda x: (1 + 2 * np.pi**2) * ez_fun(x)
mesh = self.M
ez_ana = ez_fun(mesh.cell_centers)
q_ana = q_fun(mesh.cell_centers)
ez_bc = ez_fun(mesh.boundary_edges)
MeI = mesh.get_edge_inner_product(invert_matrix=True)
M_be = mesh.boundary_edge_vector_integral
V = utils.sdiag(mesh.cell_volumes)
C = mesh.edge_curl
A = V @ C @ MeI @ C.T @ V + V
rhs = V @ q_ana + V @ C @ MeI @ M_be @ ez_bc
ez_test = Pardiso(A) * rhs
if self._meshType == "rotateCurv":
err = np.linalg.norm(mesh.cell_volumes * (ez_test - ez_ana))
else:
err = np.linalg.norm(ez_test - ez_ana) / np.sqrt(mesh.n_cells)
return err
def getError(self):
# Test function
phi = lambda x: np.cos(np.pi * x[:, 0]) * np.cos(np.pi * x[:, 1])
j_funX = lambda x: -np.pi * np.sin(np.pi * x[:, 0]) * np.cos(np.pi *
x[:, 1])
j_funY = lambda x: -np.pi * np.cos(np.pi * x[:, 0]) * np.sin(np.pi *
x[:, 1])
q_fun = lambda x: -2 * (np.pi**2) * phi(x)
mesh = self.M
phi_ana = phi(mesh.cell_centers)
q_ana = q_fun(mesh.cell_centers)
phi_bc = phi(mesh.boundary_faces)
MfI = mesh.get_face_inner_product(invert_matrix=True)
M_bf = mesh.boundary_face_scalar_integral
V = utils.sdiag(mesh.cell_volumes)
G = -mesh.face_divergence.T * V
D = mesh.face_divergence
# Sinc the xc_bc is a known, move it to the RHS!
A = V @ D @ MfI @ G
rhs = V @ q_ana - V @ D @ MfI @ M_bf @ phi_bc
phi_test = Solver(A) * rhs
if self._meshType == "rotateCurv":
err = np.linalg.norm(mesh.cell_volumes * (phi_test - phi_ana))
else:
err = np.linalg.norm(phi_test - phi_ana) / np.sqrt(mesh.n_cells)
return err
def getError(self):
# Test function
phi = lambda x: np.sin(np.pi * x[:, 0]) * np.sin(
np.pi * x[:, 1]) * np.sin(np.pi * x[:, 2])
j_funX = lambda x: np.pi * np.cos(np.pi * x[:, 0]) * np.sin(
np.pi * x[:, 1]) * np.sin(np.pi * x[:, 2])
j_funY = lambda x: np.pi * np.sin(np.pi * x[:, 0]) * np.cos(
np.pi * x[:, 1]) * np.sin(np.pi * x[:, 2])
j_funZ = lambda x: np.pi * np.sin(np.pi * x[:, 0]) * np.sin(
np.pi * x[:, 1]) * np.cos(np.pi * x[:, 2])
q_fun = lambda x: -3 * (np.pi**2) * phi(x)
mesh = self.M
phi_ana = phi(mesh.cell_centers)
q_ana = q_fun(mesh.cell_centers)
phi_bc = phi(mesh.boundary_faces)
jx_bc = j_funX(mesh.boundary_faces)
jy_bc = j_funY(mesh.boundary_faces)
jz_bc = j_funZ(mesh.boundary_faces)
j_bc = np.c_[jx_bc, jy_bc, jz_bc]
j_bc_dot_n = np.sum(j_bc * mesh.boundary_face_outward_normals, axis=-1)
MfI = mesh.get_face_inner_product(invert_matrix=True)
V = utils.sdiag(mesh.cell_volumes)
G = mesh.face_divergence.T * V
D = mesh.face_divergence
# construct matrix with robin operator
# get indices of x0 boundary, y0, and z0 boundary
n_bounary_faces = len(j_bc_dot_n)
dirichlet_locs = np.any(mesh.boundary_faces == 0.0, axis=1)
alpha = np.zeros(n_bounary_faces)
alpha[dirichlet_locs] = 1.0
beta = np.zeros(n_bounary_faces)
beta[~dirichlet_locs] = 1.0
gamma = alpha * phi_bc + beta * j_bc_dot_n
B_bc, b_bc = mesh.cell_gradient_weak_form_robin(alpha=alpha,
beta=beta,
gamma=gamma)
A = V @ D @ MfI @ (-G + B_bc)
rhs = V @ q_ana - V @ D @ MfI @ b_bc
phi_test = Pardiso(A) * rhs
if self._meshType == "rotateCurv":
err = np.linalg.norm(mesh.cell_volumes * (phi_test - phi_ana))
else:
err = np.linalg.norm(phi_test - phi_ana) / np.sqrt(mesh.n_cells)
return err
def edgeCurl(self):
"""
Construct the 3D curl operator.
"""
L = self.edge # Compute lengths of cell edges
S = self.area # Compute areas of cell faces
if getattr(self, '_edgeCurl', None) is None:
assert self.dim > 1, "Edge Curl only programed for 2 or 3D."
if self.dim == 2:
self._edgeCurl = self._edgeCurlStencil*sdiag(1/S)
elif self.dim == 3:
self._edgeCurl = sdiag(1/S)*(self._edgeCurlStencil*sdiag(L))
return self._edgeCurl
def _nodalLaplaciany(self):
Hy = sdiag(1. / self.hy)
if self.dim == 1:
return None
elif self.dim == 2:
Hy = sp.kron(Hy, speye(self.nNx))
elif self.dim == 3:
Hy = kron3(speye(self.nNz), Hy, speye(self.nNx))
return Hy.T * self._nodalGradStencily * Hy
def nodalGrad(self):
"""
Construct gradient operator (nodes to edges).
"""
if getattr(self, '_nodalGrad', None) is None:
G = self._nodalGradStencil
L = self.edge
self._nodalGrad = sdiag(1 / L) * G
return self._nodalGrad
def innerProductDeriv(v=None):
if v is None:
warnings.warn(
"Depreciation Warning: TensorMesh.innerProductDeriv."
" You should be supplying a vector. "
"Use: sdiag(u)*dMdprop", FutureWarning
)
return dMdprop
return utils.sdiag(v) * dMdprop
def cellGrad(self):
"""
The cell centered Gradient, takes you to cell faces.
"""
if getattr(self, '_cellGrad', None) is None:
G = self._cellGradStencil
S = self.area # Compute areas of cell faces & volumes
V = self.aveCC2F * self.vol # Average volume between adjacent cells
self._cellGrad = sdiag(S / V) * G
return self._cellGrad
def cellGrady(self):
if self.dim < 2:
return None
if getattr(self, '_cellGrady', None) is None:
G2 = self._cellGradyStencil
# Compute areas of cell faces & volumes
V = self.aveCC2F * self.vol
L = self.r(self.area / V, 'F', 'Fy', 'V')
self._cellGrady = sdiag(L) * G2
return self._cellGrady
def test_mat_shape(self):
o = Identity()
S = sdiag(np.r_[2, 3])[:1, :]
self.assertRaises(ValueError, lambda: S + o)
def check(exp, ans):
assert np.all((exp).todense() == ans)
check(S * o, [[2, 0]])
check(S * -o, [[-2, 0]])
def getError(self):
#Test function
phi = lambda x: np.cos(np.pi * x[:, 0]) * np.cos(np.pi * x[:, 1])
j_funX = lambda x: -np.pi * np.sin(np.pi * x[:, 0]) * np.cos(np.pi *
x[:, 1])
j_funY = lambda x: -np.pi * np.cos(np.pi * x[:, 0]) * np.sin(np.pi *
x[:, 1])
q_fun = lambda x: -2 * (np.pi**2) * phi(x)
xc_ana = phi(self.M.gridCC)
q_ana = q_fun(self.M.gridCC)
jX_ana = j_funX(self.M.gridFx)
jY_ana = j_funY(self.M.gridFy)
j_ana = np.r_[jX_ana, jY_ana]
#TODO: Check where our boundary conditions are CCx or Nx
# fxm,fxp,fym,fyp = self.M.faceBoundaryInd
# gBFx = self.M.gridFx[(fxm|fxp),:]
# gBFy = self.M.gridFy[(fym|fyp),:]
fxm, fxp, fym, fyp = self.M.cellBoundaryInd
gBFx = self.M.gridCC[(fxm | fxp), :]
gBFy = self.M.gridCC[(fym | fyp), :]
bc = phi(np.r_[gBFx, gBFy])
# P = sp.csr_matrix(([-1,1],([0,self.M.nF-1],[0,1])), shape=(self.M.nF, 2))
P, Pin, Pout = self.M.getBCProjWF('dirichlet')
Mc = self.M.getFaceInnerProduct()
McI = utils.sdInv(self.M.getFaceInnerProduct())
G = -self.M.faceDiv.T * utils.sdiag(self.M.vol)
D = self.M.faceDiv
j = McI * (G * xc_ana + P * bc)
q = D * j
# self.M.plotImage(j, 'FxFy', show_it=True)
# Rearrange if we know q to solve for x
A = D * McI * G
rhs = q_ana - D * McI * P * bc
if self.myTest == 'j':
err = np.linalg.norm((j - j_ana), np.inf)
elif self.myTest == 'q':
err = np.linalg.norm((q - q_ana), np.inf)
elif self.myTest == 'xc':
xc = Solver(A) * (rhs)
err = np.linalg.norm((xc - xc_ana), np.inf)
elif self.myTest == 'xcJ':
xc = Solver(A) * (rhs)
j = McI * (G * xc + P * bc)
err = np.linalg.norm((j - j_ana), np.inf)
return err
def cellGradx(self):
"""
Cell centered Gradient in the x dimension. Has neumann boundary
conditions.
"""
if getattr(self, '_cellGradx', None) is None:
G1 = self._cellGradxStencil
# Compute areas of cell faces & volumes
V = self.aveCC2F * self.vol
L = self.r(self.area / V, 'F', 'Fx', 'V')
self._cellGradx = sdiag(L) * G1
return self._cellGradx
def getError(self):
#Test function
phi = lambda x: np.cos(0.5 * np.pi * x)
j_fun = lambda x: -0.5 * np.pi * np.sin(0.5 * np.pi * x)
q_fun = lambda x: -0.25 * (np.pi**2) * np.cos(0.5 * np.pi * x)
xc_ana = phi(self.M.gridCC)
q_ana = q_fun(self.M.gridCC)
j_ana = j_fun(self.M.gridFx)
#TODO: Check where our boundary conditions are CCx or Nx
vecN = self.M.vectorNx
vecC = self.M.vectorCCx
phi_bc = phi(vecC[[0, -1]])
j_bc = j_fun(vecN[[0, -1]])
P, Pin, Pout = self.M.getBCProjWF([['dirichlet', 'neumann']])
Mc = self.M.getFaceInnerProduct()
McI = utils.sdInv(self.M.getFaceInnerProduct())
V = utils.sdiag(self.M.vol)
G = -Pin.T * Pin * self.M.faceDiv.T * V
D = self.M.faceDiv
j = McI * (G * xc_ana + P * phi_bc)
q = V * D * Pin.T * Pin * j + V * D * Pout.T * j_bc
# Rearrange if we know q to solve for x
A = V * D * Pin.T * Pin * McI * G
rhs = V * q_ana - V * D * Pin.T * Pin * McI * P * phi_bc - V * D * Pout.T * j_bc
# A = D*McI*G
# rhs = q_ana - D*McI*P*phi_bc
if self.myTest == 'j':
err = np.linalg.norm((Pin * j - Pin * j_ana), np.inf)
elif self.myTest == 'q':
err = np.linalg.norm((q - V * q_ana), np.inf)
elif self.myTest == 'xc':
#TODO: fix the null space
xc, info = sp.linalg.minres(A, rhs, tol=1e-6)
err = np.linalg.norm((xc - xc_ana), np.inf)
if info > 0:
print('Solve does not work well')
print('ACCURACY', np.linalg.norm(utils.mkvc(A * xc) - rhs))
elif self.myTest == 'xcJ':
#TODO: fix the null space
xc, info = sp.linalg.minres(A, rhs, tol=1e-6)
j = McI * (G * xc + P * phi_bc)
err = np.linalg.norm((Pin * j - Pin * j_ana), np.inf)
if info > 0:
print('Solve does not work well')
print('ACCURACY', np.linalg.norm(utils.mkvc(A * xc) - rhs))
return err
def getError(self):
#Test function
phi = lambda x: np.cos(np.pi * x)
j_fun = lambda x: -np.pi * np.sin(np.pi * x)
q_fun = lambda x: -(np.pi**2) * np.cos(np.pi * x)
xc_ana = phi(self.M.gridCC)
q_ana = q_fun(self.M.gridCC)
j_ana = j_fun(self.M.gridFx)
#TODO: Check where our boundary conditions are CCx or Nx
# vec = self.M.vectorNx
vec = self.M.vectorCCx
phi_bc = phi(vec[[0, -1]])
j_bc = j_fun(vec[[0, -1]])
P, Pin, Pout = self.M.getBCProjWF([['dirichlet', 'dirichlet']])
Mc = self.M.getFaceInnerProduct()
McI = utils.sdInv(self.M.getFaceInnerProduct())
V = utils.sdiag(self.M.vol)
G = -Pin.T * Pin * self.M.faceDiv.T * V
D = self.M.faceDiv
j = McI * (G * xc_ana + P * phi_bc)
q = V * D * Pin.T * Pin * j + V * D * Pout.T * j_bc
# Rearrange if we know q to solve for x
A = V * D * Pin.T * Pin * McI * G
rhs = V * q_ana - V * D * Pin.T * Pin * McI * P * phi_bc - V * D * Pout.T * j_bc
# A = D*McI*G
# rhs = q_ana - D*McI*P*phi_bc
if self.myTest == 'j':
err = np.linalg.norm((j - j_ana), np.inf)
elif self.myTest == 'q':
err = np.linalg.norm((q - V * q_ana), np.inf)
elif self.myTest == 'xc':
#TODO: fix the null space
solver = SolverCG(A, maxiter=1000)
xc = solver * (rhs)
print('ACCURACY', np.linalg.norm(utils.mkvc(A * xc) - rhs))
err = np.linalg.norm((xc - xc_ana), np.inf)
elif self.myTest == 'xcJ':
#TODO: fix the null space
xc = Solver(A) * (rhs)
print(np.linalg.norm(utils.mkvc(A * xc) - rhs))
j = McI * (G * xc + P * phi_bc)
err = np.linalg.norm((j - j_ana), np.inf)
return err
def _getInnerProductProjectionMatrices(self, projection_type, tensorType):
"""
Parameters
----------
projection_type : str
'F' for faces 'E' for edges
tensorType : TensorType
type of the tensor: TensorType(mesh, sigma)
"""
if not isinstance(tensorType, TensorType):
raise TypeError("tensorType must be an instance of TensorType.")
if projection_type not in ["F", "E"]:
raise TypeError(
"projection_type must be 'F' for faces or 'E' for edges")
d = self.dim
# We will multiply by sqrt on each side to keep symmetry
V = sp.kron(sp.identity(d),
sdiag(np.sqrt((2**(-d)) * self.cell_volumes)))
nodes = ["000", "100", "010", "110", "001", "101", "011", "111"][:2**d]
if projection_type == "F":
locs = {
"000": [("fXm", ), ("fXm", "fYm"), ("fXm", "fYm", "fZm")],
"100": [("fXp", ), ("fXp", "fYm"), ("fXp", "fYm", "fZm")],
"010": [None, ("fXm", "fYp"), ("fXm", "fYp", "fZm")],
"110": [None, ("fXp", "fYp"), ("fXp", "fYp", "fZm")],
"001": [None, None, ("fXm", "fYm", "fZp")],
"101": [None, None, ("fXp", "fYm", "fZp")],
"011": [None, None, ("fXm", "fYp", "fZp")],
"111": [None, None, ("fXp", "fYp", "fZp")],
}
proj = getattr(self, "_getFaceP" + ("x" * d))()
elif projection_type == "E":
locs = {
"000": [("eX0", ), ("eX0", "eY0"), ("eX0", "eY0", "eZ0")],
"100": [("eX0", ), ("eX0", "eY1"), ("eX0", "eY1", "eZ1")],
"010": [None, ("eX1", "eY0"), ("eX1", "eY0", "eZ2")],
"110": [None, ("eX1", "eY1"), ("eX1", "eY1", "eZ3")],
"001": [None, None, ("eX2", "eY2", "eZ0")],
"101": [None, None, ("eX2", "eY3", "eZ1")],
"011": [None, None, ("eX3", "eY2", "eZ2")],
"111": [None, None, ("eX3", "eY3", "eZ3")],
}
proj = getattr(self, "_getEdgeP" + ("x" * d))()
return [V * proj(*locs[node][d - 1]) for node in nodes]
def _getInnerProductProjectionMatrices(self, projType, tensorType):
"""
Parameters
----------
projType : str
'F' for faces 'E' for edges
tensorType : TensorType
type of the tensor: TensorType(mesh, sigma)
"""
assert isinstance(
tensorType,
TensorType), 'tensorType must be an instance of TensorType.'
assert projType in [
'F', 'E'
], "projType must be 'F' for faces or 'E' for edges"
d = self.dim
# We will multiply by sqrt on each side to keep symmetry
V = sp.kron(sp.identity(d), sdiag(np.sqrt((2**(-d)) * self.vol)))
nodes = ['000', '100', '010', '110', '001', '101', '011', '111'][:2**d]
if projType == 'F':
locs = {
'000': [('fXm', ), ('fXm', 'fYm'), ('fXm', 'fYm', 'fZm')],
'100': [('fXp', ), ('fXp', 'fYm'), ('fXp', 'fYm', 'fZm')],
'010': [None, ('fXm', 'fYp'), ('fXm', 'fYp', 'fZm')],
'110': [None, ('fXp', 'fYp'), ('fXp', 'fYp', 'fZm')],
'001': [None, None, ('fXm', 'fYm', 'fZp')],
'101': [None, None, ('fXp', 'fYm', 'fZp')],
'011': [None, None, ('fXm', 'fYp', 'fZp')],
'111': [None, None, ('fXp', 'fYp', 'fZp')]
}
proj = getattr(self, '_getFaceP' + ('x' * d))()
elif projType == 'E':
locs = {
'000': [('eX0', ), ('eX0', 'eY0'), ('eX0', 'eY0', 'eZ0')],
'100': [('eX0', ), ('eX0', 'eY1'), ('eX0', 'eY1', 'eZ1')],
'010': [None, ('eX1', 'eY0'), ('eX1', 'eY0', 'eZ2')],
'110': [None, ('eX1', 'eY1'), ('eX1', 'eY1', 'eZ3')],
'001': [None, None, ('eX2', 'eY2', 'eZ0')],
'101': [None, None, ('eX2', 'eY3', 'eZ1')],
'011': [None, None, ('eX3', 'eY2', 'eZ2')],
'111': [None, None, ('eX3', 'eY3', 'eZ3')]
}
proj = getattr(self, '_getEdgeP' + ('x' * d))()
return [V * proj(*locs[node][d - 1]) for node in nodes]
def getError(self):
# Test function
phi = lambda x: np.sin(np.pi * x[:, 0]) * np.sin(np.pi * x[:, 1])
j_funX = lambda x: np.pi * np.cos(np.pi * x[:, 0]) * np.sin(np.pi *
x[:, 1])
j_funY = lambda x: np.pi * np.sin(np.pi * x[:, 0]) * np.cos(np.pi *
x[:, 1])
q_fun = lambda x: -2 * (np.pi**2) * phi(x)
mesh = self.M
phi_ana = phi(mesh.cell_centers)
q_ana = q_fun(mesh.cell_centers)
phi_bc = phi(mesh.boundary_faces)
jx_bc = j_funX(mesh.boundary_faces)
jy_bc = j_funY(mesh.boundary_faces)
j_bc = np.c_[jx_bc, jy_bc]
j_bc_dot_n = np.sum(j_bc * mesh.boundary_face_outward_normals, axis=-1)
MfI = mesh.get_face_inner_product(invert_matrix=True)
V = utils.sdiag(mesh.cell_volumes)
G = mesh.face_divergence.T * V
D = mesh.face_divergence
# construct matrix with robin operator
alpha = 0.0
beta = 1.0
gamma = alpha * phi_bc + beta * j_bc_dot_n
B_bc, b_bc = mesh.cell_gradient_weak_form_robin(alpha=alpha,
beta=beta,
gamma=gamma)
A = V @ D @ MfI @ (-G + B_bc)
rhs = V @ q_ana - V @ D @ MfI @ b_bc
phi_test, info = linalg.minres(A, rhs, tol=1e-6)
phi_test -= phi_test.mean()
phi_ana -= phi_ana.mean()
# err = np.linalg.norm(phi_test - phi_ana)/np.sqrt(mesh.n_cells)
if self._meshType == "rotateCurv":
err = np.linalg.norm(mesh.cell_volumes * (phi_test - phi_ana))
else:
err = np.linalg.norm((phi_test - phi_ana), np.inf)
return err
def getError(self):
# Test function
phi = lambda x: np.sin(np.pi * x)
j_fun = lambda x: np.pi * np.cos(np.pi * x)
q_fun = lambda x: -(np.pi**2) * np.sin(np.pi * x)
mesh = self.M
xc_ana = phi(mesh.cell_centers)
q_ana = q_fun(mesh.cell_centers)
j_ana = j_fun(mesh.faces_x)
phi_bc = phi(mesh.boundary_faces)
j_bc = j_fun(mesh.boundary_faces)
MfI = mesh.get_face_inner_product(invert_matrix=True)
V = utils.sdiag(mesh.cell_volumes)
G = mesh.face_divergence.T * V
D = mesh.face_divergence
# construct matrix with robin operator
alpha = 0.0
beta = 1.0
gamma = alpha * phi_bc + beta * j_bc * mesh.boundary_face_outward_normals
B_bc, b_bc = mesh.cell_gradient_weak_form_robin(alpha=alpha,
beta=beta,
gamma=gamma)
j = MfI @ ((-G + B_bc) @ xc_ana + b_bc)
q = D @ j
# Since the xc_bc is a known, move it to the RHS!
A = V @ D @ MfI @ (-G + B_bc)
rhs = V @ q_ana - V @ D @ MfI @ b_bc
xc_disc, info = sp.linalg.minres(A, rhs, tol=1e-6)
if self.myTest == "j":
err = np.linalg.norm((j - j_ana), np.inf)
elif self.myTest == "xcJ":
# TODO: fix the null space
xc, info = linalg.minres(A, rhs, tol=1e-6)
j = MfI @ ((-G + B_bc) @ xc + b_bc)
err = np.linalg.norm((j - j_ana)) / np.sqrt(mesh.n_edges)
if info > 0:
print("Solve does not work well")
print("ACCURACY", np.linalg.norm(utils.mkvc(A * xc) - rhs))
return err
def getError(self):
# Test function
phi = lambda x: np.cos(0.5 * np.pi * x)
j_fun = lambda x: -0.5 * np.pi * np.sin(0.5 * np.pi * x)
q_fun = lambda x: -0.25 * (np.pi**2) * np.cos(0.5 * np.pi * x)
mesh = self.M
xc_ana = phi(mesh.cell_centers)
q_ana = q_fun(mesh.cell_centers)
j_ana = j_fun(mesh.faces_x)
phi_bc = phi(mesh.boundary_faces)
j_bc = j_fun(mesh.boundary_faces)
MfI = mesh.get_face_inner_product(invert_matrix=True)
V = utils.sdiag(mesh.cell_volumes)
G = mesh.face_divergence.T * V
D = mesh.face_divergence
# construct matrix with robin operator
alpha = np.r_[1.0, 0.0]
beta = np.r_[0.0, 1.0]
gamma = alpha * phi_bc + beta * j_bc * mesh.boundary_face_outward_normals
B_bc, b_bc = mesh.cell_gradient_weak_form_robin(alpha=alpha,
beta=beta,
gamma=gamma)
A = V @ D @ MfI @ (-G + B_bc)
rhs = V @ q_ana - V @ D @ MfI @ b_bc
if self.myTest == "xc":
xc = Solver(A) * rhs
err = np.linalg.norm(xc - xc_ana) / np.sqrt(mesh.n_cells)
elif self.myTest == "xcJ":
xc = Solver(A) * rhs
j = MfI @ ((-G + B_bc) @ xc + b_bc)
err = np.linalg.norm(j - j_ana, np.inf)
return err
def test_mat_one(self):
o = Identity()
S = sdiag(np.r_[2, 3])
def check(exp, ans):
assert np.all((exp).todense() == ans)
check(S * o, [[2, 0], [0, 3]])
check(o * S, [[2, 0], [0, 3]])
check(S * -o, [[-2, 0], [0, -3]])
check(-o * S, [[-2, 0], [0, -3]])
check(S / o, [[2, 0], [0, 3]])
check(S / -o, [[-2, 0], [0, -3]])
self.assertRaises(NotImplementedError, lambda: o / S)
check(S + o, [[3, 0], [0, 4]])
check(o + S, [[3, 0], [0, 4]])
check(S - o, [[1, 0], [0, 2]])
check(S + -o, [[1, 0], [0, 2]])
check(-o + S, [[1, 0], [0, 2]])
def simpleFail(x):
return np.sin(x), -sdiag(np.cos(x))
def simpleFunction(x):
return np.sin(x), lambda xi: sdiag(np.cos(x)) * xi
def simplePass(x):
return np.sin(x), sdiag(np.cos(x))