def test_upwind_2d_simplex_surf_discharge_negative(self):
g = simplex.StructuredTriangleGrid([2, 1], [1, 1])
R = cg.rot(-np.pi / 5., [1, 1, -1])
g.nodes = np.dot(R, g.nodes)
g.compute_geometry(is_embedded=True)
solver = upwind.Upwind()
param = Parameters(g)
dis = solver.discharge(g, np.dot(R, [-1, 0, 0]))
bf = g.tags['domain_boundary_faces'].nonzero()[0]
bc = BoundaryCondition(g, bf, bf.size * ['neu'])
param.set_bc(solver, bc)
data = {'param': param, 'discharge': dis}
M = solver.matrix_rhs(g, data)[0].todense()
deltaT = solver.cfl(g, data)
M_known = np.array([[1, 0, 0, -1],
[-1, 0, 0, 0],
[0, 0, 1, 0],
[0, 0, -1, 1]])
deltaT_known = 1 / 6
rtol = 1e-15
atol = rtol
assert np.allclose(M, M_known, rtol, atol)
assert np.allclose(deltaT, deltaT_known, rtol, atol)
def test_upwind_1d_discharge_negative_bc_neu(self):
g = structured.CartGrid(3, 1)
g.compute_geometry()
solver = upwind.Upwind()
param = Parameters(g)
dis = solver.discharge(g, [-2, 0, 0])
bf = g.tags['domain_boundary_faces'].nonzero()[0]
bc = BoundaryCondition(g, bf, bf.size * ['neu'])
bc_val = np.array([2, 0, 0, -2]).ravel('F')
param.set_bc(solver, bc)
param.set_bc_val(solver, bc_val)
data = {'param': param, 'discharge': dis}
M, rhs = solver.matrix_rhs(g, data)
deltaT = solver.cfl(g, data)
M_known = np.array([[0, -2, 0],
[0, 2, -2],
[0, 0, 2]])
rhs_known = np.array([-2, 0, 2])
deltaT_known = 1 / 12
rtol = 1e-15
atol = rtol
assert np.allclose(M.todense(), M_known, rtol, atol)
assert np.allclose(rhs, rhs_known, rtol, atol)
assert np.allclose(deltaT, deltaT_known, rtol, atol)
def test_upwind_2d_simplex_surf_discharge_positive(self):
g = simplex.StructuredTriangleGrid([2, 1], [1, 1])
R = cg.rot(np.pi / 2., [1, 1, 0])
g.nodes = np.dot(R, g.nodes)
g.compute_geometry()
solver = upwind.Upwind()
param = Parameters(g)
dis = solver.discharge(g, np.dot(R, [1, 0, 0]))
bf = g.tags["domain_boundary_faces"].nonzero()[0]
bc = BoundaryCondition(g, bf, bf.size * ["neu"])
param.set_bc(solver, bc)
data = {"param": param, "discharge": dis}
M = solver.matrix_rhs(g, data)[0].todense()
deltaT = solver.cfl(g, data)
M_known = np.array([[1, -1, 0, 0], [0, 1, 0, 0], [0, 0, 0, -1],
[-1, 0, 0, 1]])
deltaT_known = 1 / 6
rtol = 1e-15
atol = rtol
self.assertTrue(np.allclose(M, M_known, rtol, atol))
self.assertTrue(np.allclose(deltaT, deltaT_known, rtol, atol))
def test_upwind_2d_cart_surf_discharge_negative(self):
g = structured.CartGrid([3, 2], [1, 1])
R = cg.rot(np.pi/6., [1,1,0])
g.nodes = np.dot(R, g.nodes)
g.compute_geometry(is_embedded=True)
solver = upwind.Upwind()
param = Parameters(g)
dis = solver.discharge(g, np.dot(R, [-1, 0, 0]))
bf = g.get_boundary_faces()
bc = BoundaryCondition(g, bf, bf.size * ['neu'])
param.set_bc(solver, bc)
data = {'param': param, 'discharge': dis}
M = solver.matrix_rhs(g, data)[0].todense()
deltaT = solver.cfl(g, data)
M_known = 0.5 * np.array([[ 0,-1, 0, 0, 0, 0],
[ 0, 1,-1, 0, 0, 0],
[ 0, 0, 1, 0, 0, 0],
[ 0, 0, 0, 0,-1, 0],
[ 0, 0, 0, 0, 1,-1],
[ 0, 0, 0, 0, 0, 1]])
deltaT_known = 1/6
rtol = 1e-15
atol = rtol
assert np.allclose(M, M_known, rtol, atol)
assert np.allclose(deltaT, deltaT_known, rtol, atol)
def test_upwind_1d_discharge_negative_bc_dir(self):
g = structured.CartGrid(3, 1)
g.compute_geometry()
solver = upwind.Upwind()
param = Parameters(g)
dis = solver.discharge(g, [-2, 0, 0])
bf = g.tags["domain_boundary_faces"].nonzero()[0]
bc = BoundaryCondition(g, bf, bf.size * ["dir"])
bc_val = 3 * np.ones(g.num_faces).ravel("F")
param.set_bc(solver, bc)
param.set_bc_val(solver, bc_val)
data = {"param": param, "discharge": dis}
M, rhs = solver.matrix_rhs(g, data)
deltaT = solver.cfl(g, data)
M_known = np.array([[2, -2, 0], [0, 2, -2], [0, 0, 2]])
rhs_known = np.array([0, 0, 6])
deltaT_known = 1 / 12
rtol = 1e-15
atol = rtol
self.assertTrue(np.allclose(M.todense(), M_known, rtol, atol))
self.assertTrue(np.allclose(rhs, rhs_known, rtol, atol))
self.assertTrue(np.allclose(deltaT, deltaT_known, rtol, atol))
def test_upwind_3d_cart_discharge_positive(self):
g = structured.CartGrid([2, 2, 2], [1, 1, 1])
g.compute_geometry()
solver = upwind.Upwind()
param = Parameters(g)
dis = solver.discharge(g, [1, 0, 0])
bf = g.tags["domain_boundary_faces"].nonzero()[0]
bc = BoundaryCondition(g, bf, bf.size * ["neu"])
param.set_bc(solver, bc)
data = {"param": param, "discharge": dis}
M = solver.matrix_rhs(g, data)[0].todense()
deltaT = solver.cfl(g, data)
M_known = 0.25 * np.array([
[1, 0, 0, 0, 0, 0, 0, 0],
[-1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, -1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, -1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, -1, 0],
])
deltaT_known = 1 / 4
rtol = 1e-15
atol = rtol
self.assertTrue(np.allclose(M, M_known, rtol, atol))
self.assertTrue(np.allclose(deltaT, deltaT_known, rtol, atol))
def test_upwind_example1(self, if_export=False):
#######################
# 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("transport")
param = Parameters(g)
dis = advect.discharge(g, [1, 0, 0])
b_faces = g.get_all_boundary_faces()
bc = BoundaryCondition(g, b_faces, ["dir"] * b_faces.size)
bc_val = np.hstack(([1], np.zeros(g.num_faces - 1)))
param.set_bc("transport", bc)
param.set_bc_val("transport", bc_val)
data = {"param": param, "discharge": dis}
data["deltaT"] = advect.cfl(g, data)
U, rhs = advect.matrix_rhs(g, data)
M, _ = mass_matrix.MassMatrix().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)
folder = "example1"
save = Exporter(g, "conc_IE", folder)
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
if if_export:
save.write_vtk({"conc": conc}, time_step=i)
if if_export:
save.write_pvd(time)
known = np.array([
0.99969927,
0.99769441,
0.99067741,
0.97352474,
0.94064879,
0.88804726,
0.81498958,
0.72453722,
0.62277832,
0.51725056,
])
assert np.allclose(conc, known)
def test_upwind_1d_discharge_negative(self):
g = structured.CartGrid(3, 1)
g.compute_geometry()
solver = upwind.Upwind()
param = Parameters(g)
dis = solver.discharge(g, [-2, 0, 0])
bf = g.get_boundary_faces()
bc = BoundaryCondition(g, bf, bf.size * ['neu'])
param.set_bc(solver, bc)
data = {'param': param, 'discharge': dis}
M = solver.matrix_rhs(g, data)[0].todense()
deltaT = solver.cfl(g, data)
M_known = np.array([[0,-2, 0],
[0, 2,-2],
[0, 0, 2]])
deltaT_known = 1/12
rtol = 1e-15
atol = rtol
assert np.allclose(M, M_known, rtol, atol)
assert np.allclose(deltaT, deltaT_known, rtol, atol)
def test_upwind_2d_cart_discharge_positive(self):
g = structured.CartGrid([3, 2], [1, 1])
g.compute_geometry()
solver = upwind.Upwind()
param = Parameters(g)
dis = solver.discharge(g, [2, 0, 0])
bf = g.tags['domain_boundary_faces'].nonzero()[0]
bc = BoundaryCondition(g, bf, bf.size * ['neu'])
param.set_bc(solver, bc)
data = {'param': param, 'discharge': dis}
M = solver.matrix_rhs(g, data)[0].todense()
deltaT = solver.cfl(g, data)
M_known = np.array([[1, 0, 0, 0, 0, 0],
[-1, 1, 0, 0, 0, 0],
[0, -1, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, -1, 1, 0],
[0, 0, 0, 0, -1, 0]])
deltaT_known = 1 / 12
rtol = 1e-15
atol = rtol
assert np.allclose(M, M_known, rtol, atol)
assert np.allclose(deltaT, deltaT_known, rtol, atol)
def test_upwind_2d_simplex_discharge_negative(self):
g = simplex.StructuredTriangleGrid([2, 1], [1, 1])
g.compute_geometry()
solver = upwind.Upwind()
param = Parameters(g)
dis = solver.discharge(g, [-1, 0, 0])
bf = g.get_boundary_faces()
bc = BoundaryCondition(g, bf, bf.size * ['neu'])
param.set_bc(solver, bc)
data = {'param': param, 'discharge': dis}
M = solver.matrix_rhs(g, data)[0].todense()
deltaT = solver.cfl(g, data)
M_known = np.array([[ 1, 0, 0,-1],
[-1, 0, 0, 0],
[ 0, 0, 1, 0],
[ 0, 0,-1, 1]])
deltaT_known = 1/6
rtol = 1e-15
atol = rtol
assert np.allclose(M, M_known, rtol, atol)
assert np.allclose(deltaT, deltaT_known, rtol, atol)
def test_upwind_example0(self, if_export=False):
#######################
# Simple 2d upwind problem with explicit Euler scheme in time
#######################
T = 1
Nx, Ny = 4, 1
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
advect = upwind.Upwind("transport")
param = Parameters(g)
dis = advect.discharge(g, [1, 0, 0])
b_faces = g.get_all_boundary_faces()
bc = BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
bc_val = np.hstack(([1], np.zeros(g.num_faces - 1)))
param.set_bc("transport", bc)
param.set_bc_val("transport", bc_val)
data = {'param': param, 'discharge': dis}
data['deltaT'] = advect.cfl(g, data)
U, rhs = advect.matrix_rhs(g, data)
OF = advect.outflow(g, data)
M, _ = mass_matrix.MassMatrix().matrix_rhs(g, data)
conc = np.zeros(g.num_cells)
M_minus_U = M - U
invM, _ = mass_matrix.InvMassMatrix().matrix_rhs(g, data)
# Loop over the time
Nt = int(T / data['deltaT'])
time = np.empty(Nt)
folder = 'example0'
production = np.zeros(Nt)
save = Exporter(g, "conc_EE", folder)
for i in np.arange(Nt):
# Update the solution
production[i] = np.sum(OF.dot(conc))
conc = invM.dot((M_minus_U).dot(conc) + rhs)
time[i] = data['deltaT'] * i
if if_export:
save.write_vtk({"conc": conc}, time_step=i)
if if_export:
save.write_pvd(time)
known = 1.09375
assert np.sum(production) == known
def add_data_transport(gb):
gb.add_node_props(['bc', 'bc_val', 'discharge', 'apertures'])
for g, d in gb:
b_faces = g.tags['domain_boundary_faces'].nonzero()[0]
index = np.nonzero(
abs(g.face_centers[:, b_faces]) == np.ones([3, b_faces.size]))[1]
b_faces = b_faces[index]
d['bc'] = bc.BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
d['bc_val'] = {'dir': np.ones(b_faces.size)}
d['apertures'] = np.ones(g.num_cells) * np.power(.01, float(g.dim < 3))
d['discharge'] = upwind.Upwind().discharge(
g, [1, 1, 2 * np.power(1000, g.dim < 2)], d['apertures'])
gb.add_edge_prop('discharge')
for e, d in gb.edges_props():
g_h = gb.sorted_nodes_of_edge(e)[1]
d['discharge'] = gb.node_prop(g_h, 'discharge')
def add_data_transport(gb):
gb.add_node_props(["bc", "bc_val", "discharge", "apertures"])
for g, d in gb:
b_faces = g.tags["domain_boundary_faces"].nonzero()[0]
index = np.nonzero(
abs(g.face_centers[:, b_faces]) == np.ones([3, b_faces.size]))[1]
b_faces = b_faces[index]
d["bc"] = bc.BoundaryCondition(g, b_faces, ["dir"] * b_faces.size)
d["bc_val"] = {"dir": np.ones(b_faces.size)}
d["apertures"] = np.ones(g.num_cells) * np.power(.01, float(g.dim < 3))
d["discharge"] = upwind.Upwind().discharge(
g, [1, 1, 2 * np.power(1000, g.dim < 2)], d["apertures"])
gb.add_edge_prop("discharge")
for e, d in gb.edges_props():
g_h = gb.sorted_nodes_of_edge(e)[1]
d["discharge"] = gb.node_prop(g_h, "discharge")
def atest_upwind_2d_1d_cross_with_elimination(self):
"""
Simplest possible elimination scenario, one 0d-grid removed. Check on upwind
matrix, rhs, solution and time step estimate. Full solution included
(as comments) for comparison purposes if test breaks.
"""
f1 = np.array([[0, 1], [.5, .5]])
f2 = np.array([[.5, .5], [0, 1]])
domain = {"xmin": 0, "ymin": 0, "xmax": 1, "ymax": 1}
mesh_size = 0.4
mesh_kwargs = {}
mesh_kwargs["mesh_size"] = {
"mode": "constant",
"value": mesh_size,
"bound_value": mesh_size,
}
gb = pp.meshing.cart_grid([f1, f2], [2, 2], **{"physdims": [1, 1]})
# gb = pp.meshing.simplex_grid( [f1, f2],domain,**mesh_kwargs)
gb.compute_geometry()
gb.assign_node_ordering()
# Enforce node orderning because of Python 3.5 and 2.7.
# Don't do it in general.
cell_centers_1 = np.array([
[7.50000000e-01, 2.500000000e-01],
[5.00000000e-01, 5.00000000e-01],
[-5.55111512e-17, 5.55111512e-17],
])
cell_centers_2 = np.array([
[5.00000000e-01, 5.00000000e-01],
[7.50000000e-01, 2.500000000e-01],
[-5.55111512e-17, 5.55111512e-17],
])
for g, d in gb:
if g.dim == 1:
if np.allclose(g.cell_centers, cell_centers_1):
d["node_number"] = 1
elif np.allclose(g.cell_centers, cell_centers_2):
d["node_number"] = 2
else:
raise ValueError("Grid not found")
tol = 1e-3
solver = pp.TpfaMixedDim()
gb.add_node_props(["param"])
a = 1e-2
for g, d in gb:
param = pp.Parameters(g)
a_dim = np.power(a, gb.dim_max() - g.dim)
aperture = np.ones(g.num_cells) * a_dim
param.set_aperture(aperture)
kxx = np.ones(g.num_cells) * np.power(1e3, g.dim < gb.dim_max())
p = pp.SecondOrderTensor(3, kxx, kyy=kxx, kzz=kxx)
param.set_tensor("flow", p)
bound_faces = g.tags["domain_boundary_faces"].nonzero()[0]
if bound_faces.size != 0:
bound_face_centers = g.face_centers[:, bound_faces]
right = bound_face_centers[0, :] > 1 - tol
left = bound_face_centers[0, :] < tol
labels = np.array(["neu"] * bound_faces.size)
labels[right] = ["dir"]
bc_val = np.zeros(g.num_faces)
bc_dir = bound_faces[right]
bc_neu = bound_faces[left]
bc_val[bc_dir] = g.face_centers[0, bc_dir]
bc_val[bc_neu] = -g.face_areas[bc_neu] * a_dim
param.set_bc("flow",
pp.BoundaryCondition(g, bound_faces, labels))
param.set_bc_val("flow", bc_val)
# Transport
bottom = bound_face_centers[1, :] < tol
top = bound_face_centers[1, :] > 1 - tol
labels = np.array(["neu"] * bound_faces.size)
labels[np.logical_or(np.logical_or(left, right),
np.logical_or(top, bottom))] = ["dir"]
bc_val = np.zeros(g.num_faces)
param.set_bc("transport",
pp.BoundaryCondition(g, bound_faces, labels))
param.set_bc_val("transport", bc_val)
else:
param.set_bc("transport",
pp.BoundaryCondition(g, np.empty(0), np.empty(0)))
param.set_bc("flow",
pp.BoundaryCondition(g, np.empty(0), np.empty(0)))
# Transport:
source = g.cell_volumes * a_dim
param.set_source("transport", source)
d["param"] = param
gb.add_edge_props("param")
for e, d in gb.edges():
g_h = gb.nodes_of_edge(e)[1]
d["param"] = pp.Parameters(g_h)
A, rhs = solver.matrix_rhs(gb)
# p = sps.linalg.spsolve(A,rhs)
_, p_red, _, _ = condensation.solve_static_condensation(A,
rhs,
gb,
dim=0)
dim_to_remove = 0
gb_r, elimination_data = gb.duplicate_without_dimension(dim_to_remove)
condensation.compute_elimination_fluxes(gb, gb_r, elimination_data)
solver.split(gb_r, "pressure", p_red)
# pp.fvutils.compute_discharges(gb)
pp.fvutils.compute_discharges(gb_r)
# ------Transport------#
advection_discr = upwind.Upwind(physics="transport")
advection_coupling_conditions = upwind.UpwindCoupling(advection_discr)
advection_coupler = coupler.Coupler(advection_discr,
advection_coupling_conditions)
U_r, rhs_u_r = advection_coupler.matrix_rhs(gb_r)
_, rhs_src_r = pp.IntegralMixedDim(
physics="transport").matrix_rhs(gb_r)
rhs_u_r = rhs_u_r + rhs_src_r
deltaT = np.amin(
gb_r.apply_function(advection_discr.cfl,
advection_coupling_conditions.cfl).data)
theta_r = sps.linalg.spsolve(U_r, rhs_u_r)
U_known, rhs_known, theta_known, deltaT_known = known_for_elimination()
tol = 1e-7
self.assertTrue(np.isclose(deltaT, deltaT_known, tol, tol))
self.assertTrue((np.amax(np.absolute(U_r - U_known))) < tol)
self.assertTrue((np.amax(np.absolute(rhs_u_r - rhs_known))) < tol)
self.assertTrue((np.amax(np.absolute(theta_r - theta_known))) < tol)
# gb = meshing.cart_grid(f_set, [Nx, Nx, Nx])
path_to_gmsh = '~/gmsh-2.16.0-Linux/bin/gmsh'
gb = meshing.simplex_grid(f_set, domain, gmsh_path=path_to_gmsh)
gb.assign_node_ordering()
################## Transport solver ##################
print("Compute global matrix and rhs for the advection problem")
gb_r, elimination_data = gb.duplicate_without_dimension(0)
condensation.compute_elimination_fluxes(gb, gb_r, elimination_data)
add_data_transport(gb)
add_data_transport(gb_r)
upwind_solver = upwind.Upwind()
upwind_cc = upwind.UpwindCoupling(upwind_solver)
coupler_solver = coupler.Coupler(upwind_solver, upwind_cc)
U, rhs = coupler_solver.matrix_rhs(gb)
U_r, rhs_r = coupler_solver.matrix_rhs(gb_r)
deltaT = np.amin([upwind_solver.cfl(g, d) for g, d in gb])
deltaT_r = np.amin([upwind_solver.cfl(g, d) for g, d in gb_r])
T = deltaT * max(Nx, Ny) * 4
gb.add_node_prop("deltaT", None, deltaT)
gb_r.add_node_prop("deltaT", None, deltaT_r)
mass_solver = mass_matrix.MassMatrix()
coupler_solver = coupler.Coupler(mass_solver)
diffusion = second_order_tensor.SecondOrderTensorTensor(g.dim, kxx)
f = np.ones(g.num_cells) * g.cell_volumes
data = {'beta_n': beta_n, 'bc': bnd, 'bc_val': bnd_val, 'k': diffusion,
'f': f}
return data
#------------------------------------------------------------------------------#
# the f is considered twice, we guess that non-homogeneous Neumann as well.
Nx = Ny = 20
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
advection = upwind.Upwind()
diffusion = tpfa.Tpfa()
# Assign parameters
data = add_data(g, advection)
U, rhs_u = advection.matrix_rhs(g, data)
D, rhs_d = diffusion.matrix_rhs(g, data)
theta = sps.linalg.spsolve(D + U, rhs_u + rhs_d)
exporter.export_vtk(g, "advection_diffusion", {"theta": theta})
def test_upwind_example2(self, if_export=False):
#######################
# Simple 2d upwind problem with explicit Euler scheme in time coupled with
# a Darcy problem
#######################
T = 2
Nx, Ny = 10, 10
folder = 'example2'
def funp_ex(pt):
return -np.sin(pt[0]) * np.sin(pt[1]) - pt[0]
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
param = Parameters(g)
# Permeability
perm = tensor.SecondOrderTensor(g.dim, kxx=np.ones(g.num_cells))
param.set_tensor("flow", perm)
# Source term
param.set_source("flow", np.zeros(g.num_cells))
# Boundaries
b_faces = g.get_all_boundary_faces()
bc = BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
bc_val = np.zeros(g.num_faces)
bc_val[b_faces] = funp_ex(g.face_centers[:, b_faces])
param.set_bc("flow", bc)
param.set_bc_val("flow", bc_val)
# Darcy solver
data = {'param': param}
solver = vem_dual.DualVEM("flow")
D_flow, b_flow = solver.matrix_rhs(g, data)
solver_source = vem_source.DualSource('flow')
D_source, b_source = solver_source.matrix_rhs(g, data)
up = sps.linalg.spsolve(D_flow + D_source, b_flow + b_source)
p, u = solver.extract_p(g, up), solver.extract_u(g, up)
P0u = solver.project_u(g, u, data)
save = Exporter(g, "darcy", folder)
if if_export:
save.write_vtk({'pressure': p, "P0u": P0u})
# Discharge
dis = u
# Boundaries
bc = BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
bc_val = np.hstack(([1], np.zeros(g.num_faces - 1)))
param.set_bc("transport", bc)
param.set_bc_val("transport", bc_val)
data = {'param': param, 'discharge': dis}
# Advect solver
advect = upwind.Upwind("transport")
U, rhs = advect.matrix_rhs(g, data)
data['deltaT'] = advect.cfl(g, data)
M, _ = mass_matrix.MassMatrix().matrix_rhs(g, data)
conc = np.zeros(g.num_cells)
M_minus_U = M - U
invM, _ = mass_matrix.InvMassMatrix().matrix_rhs(g, data)
# Loop over the time
Nt = int(T / data['deltaT'])
time = np.empty(Nt)
save.change_name("conc_darcy")
for i in np.arange(Nt):
# Update the solution
conc = invM.dot((M_minus_U).dot(conc) + rhs)
time[i] = data['deltaT'] * i
if if_export:
save.write_vtk({"conc": conc}, time_step=i)
if if_export:
save.write_pvd(time)
known = \
np.array([9.63168200e-01, 8.64054875e-01, 7.25390695e-01,
5.72228235e-01, 4.25640080e-01, 2.99387331e-01,
1.99574336e-01, 1.26276876e-01, 7.59011550e-02,
4.33431230e-02, 3.30416807e-02, 1.13058617e-01,
2.05372538e-01, 2.78382057e-01, 3.14035373e-01,
3.09920132e-01, 2.75024694e-01, 2.23163145e-01,
1.67386939e-01, 1.16897527e-01, 1.06111312e-03,
1.11951850e-02, 3.87907727e-02, 8.38516119e-02,
1.36617802e-01, 1.82773271e-01, 2.10446545e-01,
2.14651936e-01, 1.97681518e-01, 1.66549151e-01,
3.20751341e-05, 9.85780113e-04, 6.07062715e-03,
1.99393042e-02, 4.53237556e-02, 8.00799828e-02,
1.17199623e-01, 1.47761481e-01, 1.64729339e-01,
1.65390555e-01, 9.18585872e-07, 8.08267622e-05,
8.47227168e-04, 4.08879583e-03, 1.26336029e-02,
2.88705048e-02, 5.27841497e-02, 8.10459333e-02,
1.07956484e-01, 1.27665318e-01, 2.51295298e-08,
6.29844122e-06, 1.09361990e-04, 7.56743783e-04,
3.11384414e-03, 9.04446601e-03, 2.03443897e-02,
3.75208816e-02, 5.89595194e-02, 8.11457277e-02,
6.63498510e-10, 4.73075468e-07, 1.33728945e-05,
1.30243418e-04, 7.01905707e-04, 2.55272292e-03,
6.96686157e-03, 1.52290448e-02, 2.78607282e-02,
4.40402650e-02, 1.71197497e-11, 3.47118057e-08,
1.57974045e-06, 2.13489614e-05, 1.48634295e-04,
6.68104990e-04, 2.18444135e-03, 5.58646819e-03,
1.17334966e-02, 2.09744728e-02, 4.37822313e-13,
2.52373622e-09, 1.83589660e-07, 3.40553325e-06,
3.02948532e-05, 1.66504215e-04, 6.45119867e-04,
1.90731440e-03, 4.53436628e-03, 8.99977737e-03,
1.12627412e-14, 1.84486857e-10, 2.13562387e-08,
5.39492977e-07, 6.08223906e-06, 4.05535296e-05,
1.84731221e-04, 6.25871542e-04, 1.66459389e-03,
3.59980231e-03])
assert np.allclose(conc, known)