def test_uniform_displacement():
g_list = setup_grids.setup_2d()
for g in g_list:
bound_faces = np.argwhere(
np.abs(g.cell_faces).sum(axis=1).A.ravel('F') == 1)
bound = bc.BoundaryCondition(g, bound_faces.ravel(1),
['dir'] * bound_faces.size)
constit = setup_stiffness(g)
# Python inverter is most efficient for small problems
stress, bound_stress = mpsa.mpsa(g, constit, bound, inverter='python')
div = fvutils.vector_divergence(g)
a = div * stress
d_x = np.random.rand(1)
d_y = np.random.rand(1)
d_bound = np.zeros((g.dim, g.num_faces))
d_bound[0, bound.is_dir] = d_x
d_bound[1, bound.is_dir] = d_y
rhs = div * bound_stress * d_bound.ravel('F')
d = np.linalg.solve(a.todense(), rhs)
traction = stress * d - bound_stress * d_bound.ravel('F')
assert np.max(np.abs(d[::2] - d_x)) < 1e-8
assert np.max(np.abs(d[1::2] - d_y)) < 1e-8
assert np.max(np.abs(traction)) < 1e-8
def tpfa_matrix(g, perm=None, faces=None):
"""
Compute a two-point flux approximation matrix useful related to a call of
create_partition.
Parameters
----------
g: the grid
perm: (optional) permeability, the it is not given unitary tensor is assumed
faces (np.array, int): Index of faces where TPFA should be applied.
Defaults all faces in the grid.
Returns
-------
out: sparse matrix
Two-point flux approximation matrix
"""
if perm is None:
perm = second_order_tensor.SecondOrderTensor(g.dim,
np.ones(g.num_cells))
bound = bc.BoundaryCondition(g, np.empty(0), '')
trm, _ = tpfa.tpfa(g, perm, bound, faces)
div = g.cell_faces.T
return div * trm
def test_no_dynamics_2d(self):
g_list = setup_grids.setup_2d()
for g in g_list:
bound_faces = g.get_boundary_faces()
bound = bc.BoundaryCondition(g, bound_faces.ravel('F'),
['dir'] * bound_faces.size)
flux, bound_flux, div_flow = self.mpfa_discr(g, bound)
a_flow = div_flow * flux
stress, bound_stress, grad_p, div_d, \
stabilization, div_mech = self.mpsa_discr(g, bound)
a_mech = div_mech * stress
a_biot = sps.bmat([[a_mech, grad_p],
[div_d, a_flow + stabilization]])
bval_mech = np.ones(g.num_faces * g.dim)
bval_flow = np.ones(g.num_faces)
rhs = np.hstack((div_mech * bound_stress * bval_mech,
div_flow * bound_flux * bval_flow))
sol = np.linalg.solve(a_biot.todense(), rhs)
assert np.isclose(sol, np.ones(g.num_cells * (g.dim + 1))).all()
def upwind_example2(**kwargs):
#######################
# Simple 2d upwind problem with explicit Euler scheme in time coupled with
# a Darcy problem
#######################
T = 2
Nx, Ny = 10, 10
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = second_order_tensor.SecondOrderTensor(g.dim, kxx)
def funp_ex(pt):
return -np.sin(pt[0]) * np.sin(pt[1]) - pt[0]
f = np.zeros(g.num_cells)
b_faces = g.get_boundary_faces()
bnd = bc.BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
bnd_val = {'dir': funp_ex(g.face_centers[:, b_faces])}
solver = dual.DualVEM()
data = {'k': perm, 'f': f, 'bc': bnd, 'bc_val': bnd_val}
D, rhs = solver.matrix_rhs(g, data)
up = sps.linalg.spsolve(D, rhs)
beta_n = solver.extractU(g, up)
u, p = solver.extractU(g, up), solver.extractP(g, up)
P0u = solver.projectU(g, u, data)
export_vtk(g, "darcy", {"p": p, "P0u": P0u})
advect = upwind.Upwind()
bnd_val = {'dir': np.hstack(([1], np.zeros(b_faces.size - 1)))}
data = {'beta_n': beta_n, 'bc': bnd, 'bc_val': bnd_val}
U, rhs = advect.matrix_rhs(g, data)
data = {'deltaT': advect.cfl(g, data)}
M, _ = mass_matrix.Mass().matrix_rhs(g, data)
conc = np.zeros(g.num_cells)
M_minus_U = M - U
invM, _ = mass_matrix.InvMass().matrix_rhs(g, data)
# Loop over the time
Nt = int(T / data['deltaT'])
time = np.empty(Nt)
for i in np.arange(Nt):
# Update the solution
conc = invM.dot((M_minus_U).dot(conc) + rhs)
time[i] = data['deltaT'] * i
export_vtk(g, "conc_darcy", {"conc": conc}, time_step=i)
export_pvd(g, "conc_darcy", time)
def test_uniform_flow_cart_2d():
nx = np.array([13, 13])
g = structured.CartGrid(nx)
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = second_order_tensor.SecondOrderTensor(g.dim, kxx)
bound_faces = np.argwhere(np.abs(g.cell_faces).sum(axis=1).A.ravel(1) == 1)
bound = bc.BoundaryCondition(g, bound_faces, ['dir'] * bound_faces.size)
flux = tpfa.tpfa(g, perm, bound)
def test_tpfa_cart_2d():
""" Apply TPFA on Cartesian grid, should obtain Laplacian stencil. """
# Set up 3 X 3 Cartesian grid
nx = np.array([3, 3])
g = structured.CartGrid(nx)
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = second_order_tensor.SecondOrderTensor(g.dim, kxx)
bound_faces = np.array([0, 3, 12])
bound = bc.BoundaryCondition(g, bound_faces, ['dir'] * bound_faces.size)
trm, bound_flux = tpfa.tpfa(g, perm, bound)
div = g.cell_faces.T
a = div * trm
b = (div * bound_flux).A
print(b)
# Checks on interior cell
mid = 4
assert a[mid, mid] == 4
assert a[mid - 1, mid] == -1
assert a[mid + 1, mid] == -1
assert a[mid - 3, mid] == -1
assert a[mid + 3, mid] == -1
assert np.all(b[mid, :] == 0)
# The first cell should have two Dirichlet bnds
assert a[0, 0] == 6
assert a[0, 1] == -1
assert a[0, 3] == -1
assert b[0, 0] == 2
assert b[0, 12] == 2
# Cell 3 has one Dirichlet, one Neumann face
assert a[2, 2] == 4
assert a[2, 1] == -1
assert a[2, 5] == -1
assert b[2, 3] == 2
assert b[2, 14] == 1
# Cell 2 has one Neumann face
assert a[1, 1] == 3
assert a[1, 0] == -1
assert a[1, 2] == -1
assert a[1, 4] == -1
assert b[1, 13] == 1
return a
def test_uniform_strain():
g_list = setup_grids.setup_2d()
for g in g_list:
bound_faces = np.argwhere(
np.abs(g.cell_faces).sum(axis=1).A.ravel('F') == 1)
bound = bc.BoundaryCondition(g, bound_faces.ravel('F'),
['dir'] * bound_faces.size)
mu = 1
l = 1
constit = setup_stiffness(g, mu, l)
# Python inverter is most efficient for small problems
stress, bound_stress = mpsa.mpsa(g, constit, bound, inverter='python')
div = fvutils.vector_divergence(g)
a = div * stress
xc = g.cell_centers
xf = g.face_centers
gx = np.random.rand(1)
gy = np.random.rand(1)
dc_x = np.sum(xc * gx, axis=0)
dc_y = np.sum(xc * gy, axis=0)
df_x = np.sum(xf * gx, axis=0)
df_y = np.sum(xf * gy, axis=0)
d_bound = np.zeros((g.dim, g.num_faces))
d_bound[0, bound.is_dir] = df_x[bound.is_dir]
d_bound[1, bound.is_dir] = df_y[bound.is_dir]
rhs = div * bound_stress * d_bound.ravel('F')
d = np.linalg.solve(a.todense(), rhs)
traction = stress * d - bound_stress * d_bound.ravel('F')
s_xx = (2 * mu + l) * gx + l * gy
s_xy = mu * (gx + gy)
s_yx = mu * (gx + gy)
s_yy = (2 * mu + l) * gy + l * gx
n = g.face_normals
traction_ex_x = s_xx * n[0] + s_xy * n[1]
traction_ex_y = s_yx * n[0] + s_yy * n[1]
assert np.max(np.abs(d[::2] - dc_x)) < 1e-8
assert np.max(np.abs(d[1::2] - dc_y)) < 1e-8
assert np.max(np.abs(traction[::2] - traction_ex_x)) < 1e-8
assert np.max(np.abs(traction[1::2] - traction_ex_y)) < 1e-8
def __init__(self, grid, perm, stiffness, poro, bound_flow=None,
bound_mech=None, water_compr=1, water_viscosity=1, verbose=0,
inverter='cython', eta=0):
self.g = grid
self.perm = perm
self.stiffness = stiffness
self.poro = poro
# Boundaries and boundary conditions
bound_faces = grid.get_boundary_faces()
if bound_flow is None:
self.bound_flow = bc.BoundaryCondition(grid, bound_faces.ravel('F'),
['neu'] * bound_faces.size)
elif isinstance(bound_flow, str) \
and (bound_flow.lower().strip() == 'dir'
or bound_flow.lower().strip() == 'dirichlet'):
self.bound_flow = bc.BoundaryCondition(grid, bound_faces.ravel('F'),
['dir'] * bound_faces.size)
elif isinstance(bound_flow, str) \
and (bound_flow.lower().strip() == 'neu'
or bound_flow.lower().strip() == 'neumann'):
self.bound_flow = bc.BoundaryCondition(grid, bound_faces.ravel('F'),
['neu'] * bound_faces.size)
else:
self.bound_flow = bound_flow
if bound_mech is None:
self.bound_mech = bc.BoundaryCondition(grid, bound_faces.ravel('F'),
['neu'] * bound_faces.size)
elif isinstance(bound_mech, str) \
and (bound_mech.lower().strip() == 'dir'
or bound_mech.lower().strip() == 'dirichlet'):
self.bound_mech = bc.BoundaryCondition(grid, bound_faces.ravel('F'),
['dir'] * bound_faces.size)
elif isinstance(bound_mech, str) \
and (bound_mech.lower().strip() == 'neu'
or bound_mech.lower().strip() == 'neumann'):
self.bound_mech = bc.BoundaryCondition(grid, bound_faces.ravel('F'),
['neu'] * bound_faces.size)
else:
self.bound_mech = bound_mech
self.water_compr = water_compr
self.water_viscosity = water_viscosity
self.inverter = inverter
self.eta = eta
self.verbose = verbose
self.biot_alpha = 1
def upwind_example1(**kwargs):
#######################
# Simple 2d upwind problem with implicit Euler scheme in time
#######################
T = 1
Nx, Ny = 10, 1
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
advect = upwind.Upwind()
beta_n = advect.beta_n(g, [1, 0, 0])
b_faces = g.get_boundary_faces()
bnd = bc.BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
bnd_val = {'dir': np.hstack(([1], np.zeros(b_faces.size - 1)))}
data = {'beta_n': beta_n, 'bc': bnd, 'bc_val': bnd_val}
U, rhs = advect.matrix_rhs(g, data)
data = {'deltaT': 2 * advect.cfl(g, data)}
M, _ = mass_matrix.Mass().matrix_rhs(g, data)
conc = np.zeros(g.num_cells)
# Perform an LU factorization to speedup the solver
IE_solver = sps.linalg.factorized((M + U).tocsc())
# Loop over the time
Nt = int(T / data['deltaT'])
time = np.empty(Nt)
for i in np.arange(Nt):
# Update the solution
# Backward and forward substitution to solve the system
conc = IE_solver(M.dot(conc) + rhs)
time[i] = data['deltaT'] * i
export_vtk(g, "conc_IE", {"conc": conc}, time_step=i)
export_pvd(g, "conc_IE", time)
def upwind_example0(**kwargs):
#######################
# Simple 2d upwind problem with explicit Euler scheme in time
#######################
T = 1
Nx, Ny = 10, 1
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
advect = upwind.Upwind()
beta_n = advect.beta_n(g, [1, 0, 0])
b_faces = g.get_boundary_faces()
bnd = bc.BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
bnd_val = {'dir': np.hstack(([1], np.zeros(b_faces.size - 1)))}
data = {'beta_n': beta_n, 'bc': bnd, 'bc_val': bnd_val}
U, rhs = advect.matrix_rhs(g, data)
data = {'deltaT': advect.cfl(g, data)}
M, _ = mass_matrix.Mass().matrix_rhs(g, data)
conc = np.zeros(g.num_cells)
M_minus_U = M - U
invM, _ = mass_matrix.InvMass().matrix_rhs(g, data)
# Loop over the time
Nt = int(T / data['deltaT'])
time = np.empty(Nt)
for i in np.arange(Nt):
# Update the solution
conc = invM.dot((M_minus_U).dot(conc) + rhs)
time[i] = data['deltaT'] * i
export_vtk(g, "conc_EE", {"conc": conc}, time_step=i)
export_pvd(g, "conc_EE", time)