def test_upwind_2d_cart_darcy_flux_positive(self):
g = pp.CartGrid([3, 2], [1, 1])
g.compute_geometry()
solver = pp.Upwind()
dis = solver.darcy_flux(g, [2, 0, 0])
bf = g.tags["domain_boundary_faces"].nonzero()[0]
bc = pp.BoundaryCondition(g, bf, bf.size * ["neu"])
specified_parameters = {"bc": bc, "darcy_flux": dis}
data = pp.initialize_default_data(g, {}, "transport", specified_parameters)
solver.discretize(g, data)
M = solver.assemble_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
self.assertTrue(np.allclose(M, M_known, rtol, atol))
self.assertTrue(np.allclose(deltaT, deltaT_known, rtol, atol))
def test_upwind_1d_darcy_flux_negative_bc_neu(self):
g = pp.CartGrid(3, 1)
g.compute_geometry()
solver = pp.Upwind()
dis = solver.darcy_flux(g, [-2, 0, 0])
bf = g.tags["domain_boundary_faces"].nonzero()[0]
bc = pp.BoundaryCondition(g, bf, bf.size * ["neu"])
bc_val = np.array([2, 0, 0, -2]).ravel("F")
specified_parameters = {"bc": bc, "bc_values": bc_val, "darcy_flux": dis}
data = pp.initialize_default_data(g, {}, "transport", specified_parameters)
solver.discretize(g, data)
M, rhs = solver.assemble_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
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_2d_simplex_surf_darcy_flux_negative(self):
g = pp.StructuredTriangleGrid([2, 1], [1, 1])
R = pp.map_geometry.rotation_matrix(-np.pi / 5.0, [1, 1, -1])
g.nodes = np.dot(R, g.nodes)
g.compute_geometry()
solver = pp.Upwind()
dis = solver.darcy_flux(g, np.dot(R, [-1, 0, 0]))
bf = g.tags["domain_boundary_faces"].nonzero()[0]
bc = pp.BoundaryCondition(g, bf, bf.size * ["neu"])
specified_parameters = {"bc": bc, "darcy_flux": dis}
data = pp.initialize_default_data(g, {}, "transport", specified_parameters)
solver.discretize(g, data)
M = solver.assemble_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
self.assertTrue(np.allclose(M, M_known, rtol, atol))
self.assertTrue(np.allclose(deltaT, deltaT_known, rtol, atol))
def test_upwind_example_2(self, if_export=False):
#######################
# Simple 2d upwind problem with implicit Euler scheme in time
#######################
T = 1
Nx, Ny = 10, 1
g = pp.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
advect = pp.Upwind("transport")
dis = advect.darcy_flux(g, [1, 0, 0])
b_faces = g.get_all_boundary_faces()
bc = pp.BoundaryCondition(g, b_faces, ["dir"] * b_faces.size)
bc_val = np.hstack(([1], np.zeros(g.num_faces - 1)))
specified_parameters = {
"bc": bc,
"bc_values": bc_val,
"darcy_flux": dis
}
data = pp.initialize_default_data(g, {}, "transport",
specified_parameters)
time_step = advect.cfl(g, data)
data[pp.PARAMETERS]["transport"]["time_step"] = time_step
advect.discretize(g, data)
U, rhs = advect.assemble_matrix_rhs(g, data)
rhs = time_step * rhs
U = time_step * U
mass = pp.MassMatrix("transport")
mass.discretize(g, data)
M, _ = mass.assemble_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 / time_step)
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] = time_step * i
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_example_1(self, if_export=False):
#######################
# Simple 2d upwind problem with explicit Euler scheme in time
#######################
T = 1
Nx, Ny = 4, 1
g = pp.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
advect = pp.Upwind("transport")
dis = advect.darcy_flux(g, [1, 0, 0])
b_faces = g.get_all_boundary_faces()
bc = pp.BoundaryCondition(g, b_faces, ["dir"] * b_faces.size)
bc_val = np.hstack(([1], np.zeros(g.num_faces - 1)))
specified_parameters = {
"bc": bc,
"bc_values": bc_val,
"darcy_flux": dis
}
data = pp.initialize_default_data(g, {}, "transport",
specified_parameters)
time_step = advect.cfl(g, data)
data[pp.PARAMETERS]["transport"]["time_step"] = time_step
advect.discretize(g, data)
U, rhs = advect.assemble_matrix_rhs(g, data)
rhs = time_step * rhs
U = time_step * U
OF = advect.outflow(g, data)
mass = pp.MassMatrix("transport")
mass.discretize(g, data)
M, _ = mass.assemble_matrix_rhs(g, data)
conc = np.zeros(g.num_cells)
M_minus_U = M - U
inv_mass = pp.InvMassMatrix("transport")
inv_mass.discretize(g, data)
invM, _ = inv_mass.assemble_matrix_rhs(g, data)
# Loop over the time
Nt = int(T / time_step)
time = np.empty(Nt)
production = np.zeros(Nt)
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] = time_step * i
known = 1.09375
assert np.sum(production) == known
def __init__(self, keyword, grids):
if isinstance(grids, list):
self._grids = grids
else:
self._grids = [grids]
self._discretization = pp.Upwind(keyword)
self._name = "Upwind"
self.keyword = keyword
self.upwind: _MergedOperator
self.rhs: _MergedOperator
self.outflow_neumann: _MergedOperator
_wrap_discretization(self, self._discretization, grids)
def advdiff(gb, param, model_flow):
model = "transport"
model_data_adv, model_data_diff = data.advdiff(gb, model, model_flow,
param)
# discretization operator names
adv_id = "advection"
diff_id = "diffusion"
# variable names
variable = "scalar"
mortar_adv = "lambda_" + variable + "_" + adv_id
mortar_diff = "lambda_" + variable + "_" + diff_id
# discretization operatr
discr_adv = pp.Upwind(model_data_adv)
discr_diff = pp.Tpfa(model_data_diff)
coupling_adv = pp.UpwindCoupling(model_data_adv)
coupling_diff = pp.RobinCoupling(model_data_diff, discr_diff)
for g, d in gb:
d[pp.PRIMARY_VARIABLES] = {variable: {"cells": 1}}
d[pp.DISCRETIZATION] = {
variable: {
adv_id: discr_adv,
diff_id: discr_diff
}
}
for e, d in gb.edges():
g_slave, g_master = gb.nodes_of_edge(e)
d[pp.PRIMARY_VARIABLES] = {
mortar_adv: {
"cells": 1
},
mortar_diff: {
"cells": 1
}
}
d[pp.COUPLING_DISCRETIZATION] = {
adv_id: {
g_slave: (variable, adv_id),
g_master: (variable, adv_id),
e: (mortar_adv, coupling_adv)
},
diff_id: {
g_slave: (variable, diff_id),
g_master: (variable, diff_id),
e: (mortar_diff, coupling_diff)
}
}
# setup the advection-diffusion problem
assembler = pp.Assembler()
logger.info(
"Assemble the advective and diffusive terms of the transport problem")
A, b, block_dof, full_dof = assembler.assemble_matrix_rhs(gb)
logger.info("done")
# mass term
mass_id = "mass"
discr_mass = pp.MassMatrix(model_data_adv)
for g, d in gb:
d[pp.PRIMARY_VARIABLES] = {variable: {"cells": 1}}
d[pp.DISCRETIZATION] = {variable: {mass_id: discr_mass}}
gb.remove_edge_props(pp.COUPLING_DISCRETIZATION)
for e, d in gb.edges():
g_slave, g_master = gb.nodes_of_edge(e)
d[pp.PRIMARY_VARIABLES] = {
mortar_adv: {
"cells": 1
},
mortar_diff: {
"cells": 1
}
}
logger.info("Assemble the mass term of the transport problem")
M, _, _, _ = assembler.assemble_matrix_rhs(gb)
logger.info("done")
# Perform an LU factorization to speedup the solver
#IE_solver = sps.linalg.factorized((M + A).tocsc())
# time loop
logger.info("Prepare the exporting")
save = pp.Exporter(gb, "solution", folder=param["folder"])
logger.info("done")
variables = [variable, param["pressure"], param["P0_flux"]]
x = np.ones(A.shape[0]) * param["initial_advdiff"]
logger.info("Start the time loop with " + str(param["n_steps"]) + " steps")
for i in np.arange(param["n_steps"]):
#x = IE_solver(b + M.dot(x))
logger.info("Solve the linear system for time step " + str(i))
x = sps.linalg.spsolve(M + A, b + M.dot(x))
logger.info("done")
logger.info("Variable post-process")
assembler.distribute_variable(gb, x, block_dof, full_dof)
logger.info("done")
logger.info("Export variable")
save.write_vtk(variables, time_step=i)
logger.info("done")
save.write_pvd(np.arange(param["n_steps"]) * param["time_step"])
def permeability_upscaling(network,
data,
mesh_args,
directions,
do_viz=True,
tracer_transport=False):
""" Compute bulk permeabilities for a 2d domain with a fracture network.
The function sets up a flow field in the specified directions, and calculates
an upscaled permeability of corresponding to the calculated flow configuration.
Parameters:
network (pp.FractureNetwork2d): Network, with domain.
data (dictionary): Data to be specified.
mesh_args (dictionray): Parameters for meshing. See FractureNetwork2d.mesh()
for details.
directions (np.array): Directions to upscale permeabilities.
Indicated by 0 (x-direction) and or 1 (y-direction).
Returns:
np.array, dim directions.size: Upscaled permeability in each of the directions.
sim_info: Various information on the time spent on this simulation.
"""
directions = np.asarray(directions)
upscaled_perm = np.zeros(directions.size)
sim_info = {}
for di, direct in enumerate(directions):
tic = time.time()
filename = str(uuid.uuid4())
gb = network.mesh(tol=network.tol,
mesh_args=mesh_args,
file_name=filename)
toc = time.time()
if di == 0:
sim_info["num_cells_2d"] = gb.grids_of_dimension(2)[0].num_cells
sim_info["time_for_meshing"] = toc - tic
gb = _setup_simulation_flow(gb, data, direct)
pressure_kw = "flow"
mpfa = pp.Mpfa(pressure_kw)
for g, d in gb:
d[pp.PRIMARY_VARIABLES] = {"pressure": {"cells": 1}}
d[pp.DISCRETIZATION] = {"pressure": {"diffusive": mpfa}}
coupler = pp.RobinCoupling(pressure_kw, mpfa)
for e, d in gb.edges():
g1, g2 = gb.nodes_of_edge(e)
d[pp.PRIMARY_VARIABLES] = {"mortar_darcy_flux": {"cells": 1}}
d[pp.COUPLING_DISCRETIZATION] = {
"lambda": {
g1: ("pressure", "diffusive"),
g2: ("pressure", "diffusive"),
e: ("mortar_darcy_flux", coupler),
}
}
assembler = pp.Assembler(gb)
assembler.discretize()
# Discretize
tic = time.time()
A, b = assembler.assemble_matrix_rhs()
if di == 0:
toc = time.time()
sim_info["Time_for_assembly"] = toc - tic
tic = time.time()
p = spla.spsolve(A, b)
if di == 0:
toc = time.time()
sim_info["Time_for_pressure_solve"] = toc - tic
assembler.distribute_variable(p)
# Post processing to recover flux
tot_pressure = 0
tot_area = 0
tot_inlet_flux = 0
for g, d in gb:
# Inlet faces of this grid
inlet = d.get("inlet_faces", None)
# If no inlet, no need to do anything
if inlet is None or inlet.size == 0:
continue
# Compute flux field in the grid
# Internal flux
flux = (d[pp.DISCRETIZATION_MATRICES][pressure_kw]["flux"] *
d[pp.STATE]["pressure"])
# Contribution from the boundary
bound_flux_discr = d[
pp.DISCRETIZATION_MATRICES][pressure_kw]["bound_flux"]
flux += bound_flux_discr * d["parameters"][pressure_kw]["bc_values"]
# Add contribution from all neighboring lower-dimensional interfaces
for e, d_e in gb.edges_of_node(g):
mg = d_e["mortar_grid"]
if mg.dim == g.dim - 1:
flux += (bound_flux_discr * mg.mortar_to_master_int() *
d_e[pp.STATE]["mortar_darcy_flux"])
# Add contribution to total inlet flux
tot_inlet_flux += np.abs(flux[inlet]).sum()
# Next, find the pressure at the inlet faces
# The pressure is calculated as the values in the neighboring cells,
# plus an offset from the incell-variations
pressure_cell = (d[pp.DISCRETIZATION_MATRICES][pressure_kw]
["bound_pressure_cell"] * d[pp.STATE]["pressure"])
pressure_flux = (d[pp.DISCRETIZATION_MATRICES][pressure_kw]
["bound_pressure_face"] *
d["parameters"][pressure_kw]["bc_values"])
inlet_pressure = pressure_cell[inlet] + pressure_flux[inlet]
# Scale the pressure at the face with the face length.
# Also include an aperture scaling here: For the highest dimensional grid, this
# will be as scaling with 1.
aperture = d[pp.PARAMETERS][pressure_kw]["aperture"][0]
tot_pressure += np.sum(inlet_pressure * g.face_areas[inlet] *
aperture)
# Add to the toal outlet area
tot_area += g.face_areas[inlet].sum() * aperture
# Also compute the cross sectional area of the domain
if g.dim == gb.dim_max():
# Extension of the domain in the active direction
dx = g.nodes[direct].max() - g.nodes[direct].min()
# Mean pressure at the inlet, which will also be mean pressure difference
# over the domain
mean_pressure = tot_pressure / tot_area
# The upscaled permeability
upscaled_perm[di] = (tot_inlet_flux / tot_area) * dx / mean_pressure
# End of post processing for permeability upscaling
if tracer_transport:
_setup_simulation_tracer(gb, data, direct)
tracer_variable = "temperature"
temperature_kw = "transport"
mortar_variable = "lambda_adv"
# Identifier of the two terms of the equation
adv = "advection"
advection_term = "advection"
mass_term = "mass"
advection_coupling_term = "advection_coupling"
adv_discr = pp.Upwind(temperature_kw)
adv_coupling = pp.UpwindCoupling(temperature_kw)
mass_discretization = pp.MassMatrix(temperature_kw)
for g, d in gb:
d[pp.PRIMARY_VARIABLES] = {tracer_variable: {"cells": 1}}
d[pp.DISCRETIZATION] = {
tracer_variable: {
advection_term: adv_discr,
mass_term: mass_discretization,
}
}
for e, d in gb.edges():
g1, g2 = gb.nodes_of_edge(e)
d[pp.PRIMARY_VARIABLES] = {mortar_variable: {"cells": 1}}
d[pp.COUPLING_DISCRETIZATION] = {
advection_coupling_term: {
g1: (tracer_variable, adv),
g2: (tracer_variable, adv),
e: (mortar_variable, adv_coupling),
}
}
pp.fvutils.compute_darcy_flux(gb,
keyword_store=temperature_kw,
lam_name="mortar_darcy_flux")
A, b, block_dof, full_dof = assembler.assemble_matrix_rhs(
gb,
active_variables=[tracer_variable, mortar_variable],
add_matrices=False,
)
advection_coupling_term += ("_" + mortar_variable + "_" +
tracer_variable + "_" +
tracer_variable)
mass_term += "_" + tracer_variable
advection_term += "_" + tracer_variable
lhs = A[mass_term] + data["time_step"] * (
A[advection_term] + A[advection_coupling_term])
rhs_source_adv = data["time_step"] * (b[advection_term] +
b[advection_coupling_term])
IEsolver = spla.factorized(lhs)
save_every = 10
n_steps = int(np.round(data["t_max"] / data["time_step"]))
# Initial condition
tracer = np.zeros(rhs_source_adv.size)
assembler.distribute_variable(
gb,
tracer,
block_dof,
full_dof,
variable_names=[tracer_variable, mortar_variable],
)
# Exporter
exporter = pp.Exporter(gb, name=tracer_variable, folder="viz_tmp")
export_fields = [tracer_variable]
for i in range(n_steps):
if np.isclose(i % save_every, 0):
# Export existing solution (final export is taken care of below)
assembler.distribute_variable(
gb,
tracer,
block_dof,
full_dof,
variable_names=[tracer_variable, mortar_variable],
)
exporter.write_vtk(export_fields,
time_step=int(i // save_every))
tracer = IEsolver(A[mass_term] * tracer + rhs_source_adv)
exporter.write_vtk(export_fields,
time_step=(n_steps // save_every))
time_steps = np.arange(0, data["t_max"] + data["time_step"],
save_every * data["time_step"])
exporter.write_pvd(time_steps)
return upscaled_perm, sim_info
def transport(gb, data, solver_name, folder, callback=None, save_every=1):
grid_variable = "tracer"
mortar_variable = "mortar_tracer"
kw = "transport"
# Identifier of the discretization operator on each grid
advection_term = "advection"
source_term = "source"
mass_term = "mass"
# Identifier of the discretization operator between grids
advection_coupling_term = "advection_coupling"
# Discretization objects
node_discretization = pp.Upwind(kw)
source_discretization = pp.ScalarSource(kw)
mass_discretization = pp.MassMatrix(kw)
edge_discretization = pp.UpwindCoupling(kw)
# Loop over the nodes in the GridBucket, define primary variables and discretization schemes
for g, d in gb:
# Assign primary variables on this grid. It has one degree of freedom per cell.
d[pp.PRIMARY_VARIABLES] = {grid_variable: {"cells": 1, "faces": 0}}
# Assign discretization operator for the variable.
d[pp.DISCRETIZATION] = {
grid_variable: {
advection_term: node_discretization,
source_term: source_discretization,
mass_term: mass_discretization,
}
}
# Loop over the edges in the GridBucket, define primary variables and discretizations
for e, d in gb.edges():
g1, g2 = gb.nodes_of_edge(e)
# The mortar variable has one degree of freedom per cell in the mortar grid
d[pp.PRIMARY_VARIABLES] = {mortar_variable: {"cells": 1}}
d[pp.COUPLING_DISCRETIZATION] = {
advection_coupling_term: {
g1: (grid_variable, advection_term),
g2: (grid_variable, advection_term),
e: (mortar_variable, edge_discretization),
}
}
d[pp.DISCRETIZATION_MATRICES] = {kw: {}}
assembler = pp.Assembler()
# Assemble the linear system, using the information stored in the GridBucket. By
# not adding the matrices, we can arrange them at will to obtain the efficient
# solver defined below, which LU factorizes the system only once for all time steps.
A, b, block_dof, full_dof = assembler.assemble_matrix_rhs(
gb,
active_variables=[grid_variable, mortar_variable],
add_matrices=False)
advection_coupling_term += ("_" + mortar_variable + "_" + grid_variable +
"_" + grid_variable)
mass_term += "_" + grid_variable
advection_term += "_" + grid_variable
source_term += "_" + grid_variable
lhs = A[mass_term] + data["time_step"] * (A[advection_term] +
A[advection_coupling_term])
rhs_source_adv = b[source_term] + data["time_step"] * (
b[advection_term] + b[advection_coupling_term])
IEsolver = sps.linalg.factorized(lhs)
n_steps = int(np.round(data["t_max"] / data["time_step"]))
# Initial condition
tracer = np.zeros(rhs_source_adv.size)
assembler.distribute_variable(
gb,
tracer,
block_dof,
full_dof,
variable_names=[grid_variable, mortar_variable])
# Exporter
exporter = pp.Exporter(gb, name="tracer", folder=folder)
export_fields = ["tracer"]
# Keep track of the outflow for each time step
outflow = np.empty(0)
# Time loop
for i in range(n_steps):
# Export existing solution (final export is taken care of below)
assembler.distribute_variable(
gb,
tracer,
block_dof,
full_dof,
variable_names=[grid_variable, mortar_variable],
)
if np.isclose(i % save_every, 0):
exporter.write_vtk(export_fields, time_step=int(i // save_every))
tracer = IEsolver(A[mass_term] * tracer + rhs_source_adv)
outflow = compute_flow_rate(gb, grid_variable, outflow)
if callback is not None:
callback(gb)
exporter.write_vtk(export_fields, time_step=(n_steps // save_every))
time_steps = np.arange(0, data["t_max"] + data["time_step"],
save_every * data["time_step"])
exporter.write_pvd(time_steps)
return tracer, outflow, A, b, block_dof, full_dof
def test_upwind_coupling_3d_2d_1d_0d(self):
f1 = np.array([[0, 1, 1, 0], [0, 0, 1, 1], [0.5, 0.5, 0.5, 0.5]])
f2 = np.array([[0.5, 0.5, 0.5, 0.5], [0, 1, 1, 0], [0, 0, 1, 1]])
f3 = np.array([[0, 1, 1, 0], [0.5, 0.5, 0.5, 0.5], [0, 0, 1, 1]])
gb = pp.meshing.cart_grid([f1, f2, f3], [2, 2, 2],
**{"physdims": [1, 1, 1]})
gb.compute_geometry()
gb.assign_node_ordering()
cell_centers1 = np.array([[0.25, 0.75, 0.25, 0.75],
[0.25, 0.25, 0.75, 0.75],
[0.5, 0.5, 0.5, 0.5]])
cell_centers2 = np.array([[0.5, 0.5, 0.5, 0.5],
[0.25, 0.25, 0.75, 0.75],
[0.75, 0.25, 0.75, 0.25]])
cell_centers3 = np.array([[0.25, 0.75, 0.25,
0.75], [0.5, 0.5, 0.5, 0.5],
[0.25, 0.25, 0.75, 0.75]])
cell_centers4 = np.array([[0.5], [0.25], [0.5]])
cell_centers5 = np.array([[0.5], [0.75], [0.5]])
cell_centers6 = np.array([[0.75], [0.5], [0.5]])
cell_centers7 = np.array([[0.25], [0.5], [0.5]])
cell_centers8 = np.array([[0.5], [0.5], [0.25]])
cell_centers9 = np.array([[0.5], [0.5], [0.75]])
for g, d in gb:
if np.allclose(g.cell_centers[:, 0], cell_centers1[:, 0]):
d["node_number"] = 1
elif np.allclose(g.cell_centers[:, 0], cell_centers2[:, 0]):
d["node_number"] = 2
elif np.allclose(g.cell_centers[:, 0], cell_centers3[:, 0]):
d["node_number"] = 3
elif np.allclose(g.cell_centers[:, 0], cell_centers4[:, 0]):
d["node_number"] = 4
elif np.allclose(g.cell_centers[:, 0], cell_centers5[:, 0]):
d["node_number"] = 5
elif np.allclose(g.cell_centers[:, 0], cell_centers6[:, 0]):
d["node_number"] = 6
elif np.allclose(g.cell_centers[:, 0], cell_centers7[:, 0]):
d["node_number"] = 7
elif np.allclose(g.cell_centers[:, 0], cell_centers8[:, 0]):
d["node_number"] = 8
elif np.allclose(g.cell_centers[:, 0], cell_centers9[:, 0]):
d["node_number"] = 9
else:
pass
for e, d in gb.edges():
g1, g2 = gb.nodes_of_edge(e)
n1 = gb.node_props(g1, "node_number")
n2 = gb.node_props(g2, "node_number")
if n1 == 1 and n2 == 0:
d["edge_number"] = 0
elif n1 == 2 and n2 == 0:
d["edge_number"] = 1
elif n1 == 3 and n2 == 0:
d["edge_number"] = 2
elif n1 == 4 and n2 == 1:
d["edge_number"] = 3
elif n1 == 5 and n2 == 1:
d["edge_number"] = 4
elif n1 == 6 and n2 == 1:
d["edge_number"] = 5
elif n1 == 7 and n2 == 1:
d["edge_number"] = 6
elif n1 == 4 and n2 == 2:
d["edge_number"] = 7
elif n1 == 5 and n2 == 2:
d["edge_number"] = 8
elif n1 == 8 and n2 == 2:
d["edge_number"] = 9
elif n1 == 9 and n2 == 2:
d["edge_number"] = 10
elif n1 == 6 and n2 == 3:
d["edge_number"] = 11
elif n1 == 7 and n2 == 3:
d["edge_number"] = 12
elif n1 == 8 and n2 == 3:
d["edge_number"] = 13
elif n1 == 9 and n2 == 3:
d["edge_number"] = 14
elif n1 == 10 and n2 == 4:
d["edge_number"] = 15
elif n1 == 10 and n2 == 5:
d["edge_number"] = 16
elif n1 == 10 and n2 == 6:
d["edge_number"] = 17
elif n1 == 10 and n2 == 7:
d["edge_number"] = 18
elif n1 == 10 and n2 == 8:
d["edge_number"] = 19
elif n1 == 10 and n2 == 9:
d["edge_number"] = 20
else:
raise ValueError
# define discretization
key = "transport"
upwind = pp.Upwind(key)
upwind_coupling = pp.UpwindCoupling(key)
variable = "T"
assign_discretization(gb, upwind, upwind_coupling, variable)
# assign parameters
tol = 1e-3
gb.add_node_props(["param"])
a = 1e-2
for g, d in gb:
aperture = np.ones(g.num_cells) * np.power(a, gb.dim_max() - g.dim)
specified_parameters = {"aperture": aperture}
bound_faces = g.tags["domain_boundary_faces"].nonzero()[0]
if bound_faces.size != 0:
bound_face_centers = g.face_centers[:, bound_faces]
left = bound_face_centers[0, :] < tol
right = bound_face_centers[0, :] > 1 - tol
labels = np.array(["neu"] * bound_faces.size)
labels[np.logical_or(left, right)] = ["dir"]
bc_val = np.zeros(g.num_faces)
bc_dir = bound_faces[np.logical_or(left, right)]
bc_val[bc_dir] = 1
bound = pp.BoundaryCondition(g, bound_faces, labels)
specified_parameters.update({"bc": bound, "bc_values": bc_val})
pp.initialize_default_data(g, d, "transport", specified_parameters)
add_constant_darcy_flux(gb, upwind, [1, 0, 0], a)
assembler = pp.Assembler(gb)
assembler.discretize()
U_tmp, rhs = assembler.assemble_matrix_rhs()
grids = np.empty(gb.num_graph_nodes() + gb.num_graph_edges(),
dtype=np.object)
variables = np.empty_like(grids)
for g, d in gb:
grids[d["node_number"]] = g
variables[d["node_number"]] = variable
for e, d in gb.edges():
grids[d["edge_number"] + gb.num_graph_nodes()] = e
variables[d["edge_number"] + gb.num_graph_nodes()] = "lambda_u"
U, rhs = permute_matrix_vector(U_tmp, rhs, assembler.block_dof,
assembler.full_dof, grids, variables)
theta = np.linalg.solve(U.A, rhs)
# deltaT = solver.cfl(gb)
(
U_known,
rhs_known,
) = (_helper_test_upwind_coupling.
matrix_rhs_for_test_upwind_coupling_3d_2d_1d_0d())
theta_known = np.array([
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1,
1,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
0,
0,
0,
0,
0,
0,
0,
0,
-0.25,
-0.25,
-0.25,
-0.25,
0.25,
0.25,
0.25,
0.25,
0,
0,
0,
0,
0,
0,
0,
0,
-5.0e-03,
5.0e-03,
-5.0e-03,
5.0e-03,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
-5e-03,
5e-03,
-5e-03,
5e-03,
0,
0,
-1e-04,
1e-04,
0,
0,
])
rtol = 1e-15
atol = rtol
self.assertTrue(np.allclose(U.todense(), U_known, rtol, atol))
self.assertTrue(np.allclose(rhs, rhs_known, rtol, atol))
self.assertTrue(np.allclose(theta, theta_known, rtol, atol))
def test_upwind_coupling_2d_1d_left_right(self):
gb, _ = pp.grid_buckets_2d.single_horizontal([1, 2], simplex=False)
tol = 1e-3
# define discretization
key = "transport"
upwind = pp.Upwind(key)
upwind_coupling = pp.UpwindCoupling(key)
variable = "T"
assign_discretization(gb, upwind, upwind_coupling, variable)
# assign parameters
gb.add_node_props(["param"])
a = 1e-2
for g, d in gb:
aperture = np.ones(g.num_cells) * np.power(a, gb.dim_max() - g.dim)
specified_parameters = {"aperture": aperture}
bound_faces = g.tags["domain_boundary_faces"].nonzero()[0]
if bound_faces.size != 0:
bound_face_centers = g.face_centers[:, bound_faces]
left = bound_face_centers[0, :] < tol
right = bound_face_centers[0, :] > 1 - tol
labels = np.array(["neu"] * bound_faces.size)
labels[np.logical_or(left, right)] = ["dir"]
bc_val = np.zeros(g.num_faces)
bc_dir = bound_faces[np.logical_or(left, right)]
bc_val[bc_dir] = 1
bound = pp.BoundaryCondition(g, bound_faces, labels)
specified_parameters.update({"bc": bound, "bc_values": bc_val})
pp.initialize_default_data(g, d, "transport", specified_parameters)
add_constant_darcy_flux(gb, upwind, [1, 0, 0], a)
assembler = pp.Assembler(gb)
assembler.discretize()
U_tmp, rhs = assembler.assemble_matrix_rhs()
grids = np.empty(gb.num_graph_nodes() + gb.num_graph_edges(),
dtype=np.object)
variables = np.empty_like(grids)
for g, d in gb:
grids[d["node_number"]] = g
variables[d["node_number"]] = variable
for e, d in gb.edges():
grids[d["edge_number"] + gb.num_graph_nodes()] = e
variables[d["edge_number"] + gb.num_graph_nodes()] = "lambda_u"
U, rhs = permute_matrix_vector(U_tmp, rhs, assembler.block_dof,
assembler.full_dof, grids, variables)
theta = np.linalg.solve(U.A, rhs)
# deltaT = solver.cfl(gb)
U_known = np.array([
[0.5, 0.0, 0.0, 0.0, 1.0],
[0.0, 0.5, 0.0, 1.0, 0.0],
[0.0, 0.0, 0.01, -1.0, -1.0],
[0.0, 0.0, 0.0, -1.0, 0.0],
[0.0, 0.0, 0.0, 0.0, -1.0],
])
rhs_known = np.array([0.5, 0.5, 1e-2, 0, 0])
theta_known = np.array([1, 1, 1, 0, 0])
rtol = 1e-15
atol = rtol
self.assertTrue(np.allclose(U.todense(), U_known, rtol, atol))
self.assertTrue(np.allclose(rhs, rhs_known, rtol, atol))
self.assertTrue(np.allclose(theta, theta_known, rtol, atol))
def test_upwind_coupling_2d_1d_left_right_cross(self):
gb, _ = pp.grid_buckets_2d.two_intersecting([2, 2], simplex=False)
# 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, 2.5000000e-01],
[0.50000000e00, 0.50000000e00],
[-5.55111512e-17, 5.55111512e-17],
])
cell_centers_2 = np.array([
[0.50000000e00, 0.50000000e00],
[7.50000000e-01, 2.5000000e-01],
[-5.55111512e-17, 5.55111512e-17],
])
for g, d in gb:
if g.dim == 2:
d["node_number"] = 0
elif 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")
elif g.dim == 0:
d["node_number"] = 3
else:
raise ValueError
for e, d in gb.edges():
g1, g2 = gb.nodes_of_edge(e)
n1 = gb.node_props(g1, "node_number")
n2 = gb.node_props(g2, "node_number")
if n1 == 1 and n2 == 0:
d["edge_number"] = 0
elif n1 == 2 and n2 == 0:
d["edge_number"] = 1
elif n1 == 3 and n2 == 1:
d["edge_number"] = 2
elif n1 == 3 and n2 == 2:
d["edge_number"] = 3
else:
raise ValueError
# define discretization
key = "transport"
upwind = pp.Upwind(key)
upwind_coupling = pp.UpwindCoupling(key)
variable = "T"
assign_discretization(gb, upwind, upwind_coupling, variable)
# define parameters
tol = 1e-3
gb.add_node_props(["param"])
a = 1e-2
for g, d in gb:
aperture = np.ones(g.num_cells) * np.power(a, gb.dim_max() - g.dim)
specified_parameters = {"aperture": aperture}
bound_faces = g.tags["domain_boundary_faces"].nonzero()[0]
if bound_faces.size != 0:
bound_face_centers = g.face_centers[:, bound_faces]
left = bound_face_centers[0, :] < tol
right = bound_face_centers[0, :] > 1 - tol
labels = np.array(["neu"] * bound_faces.size)
labels[np.logical_or(left, right)] = ["dir"]
bc_val = np.zeros(g.num_faces)
bc_dir = bound_faces[np.logical_or(left, right)]
bc_val[bc_dir] = 1
bound = pp.BoundaryCondition(g, bound_faces, labels)
specified_parameters.update({"bc": bound, "bc_values": bc_val})
else:
bound = pp.BoundaryCondition(g, np.empty(0), np.empty(0))
specified_parameters.update({"bc": bound})
pp.initialize_default_data(g, d, "transport", specified_parameters)
add_constant_darcy_flux(gb, upwind, [1, 0, 0], a)
assembler = pp.Assembler(gb)
assembler.discretize()
U_tmp, rhs = assembler.assemble_matrix_rhs()
grids = np.empty(gb.num_graph_nodes() + gb.num_graph_edges(),
dtype=np.object)
variables = np.empty_like(grids)
for g, d in gb:
grids[d["node_number"]] = g
variables[d["node_number"]] = variable
for e, d in gb.edges():
grids[d["edge_number"] + gb.num_graph_nodes()] = e
variables[d["edge_number"] + gb.num_graph_nodes()] = "lambda_u"
U, rhs = permute_matrix_vector(U_tmp, rhs, assembler.block_dof,
assembler.full_dof, grids, variables)
theta = np.linalg.solve(U.A, rhs)
# deltaT = solver.cfl(gb)
U_known = np.array([
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
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,
],
[
0.0,
0.5,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.5,
0.0,
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,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.01,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
1.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
-1.0,
0.0,
0.0,
1.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
-1.0,
-1.0,
-1.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-0.5,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-0.5,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.5,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.5,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.01,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-0.01,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
],
])
theta_known = np.array([
1,
1,
1,
1,
1,
1,
1,
1,
1,
0,
0,
0,
0,
-0.5,
-0.5,
0.5,
0.5,
0.01,
-0.01,
0,
0,
])
rhs_known = np.array([
0.5,
0.0,
0.5,
0.0,
0.0,
0.01,
0.0,
0.0,
0.0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
])
rtol = 1e-15
atol = rtol
self.assertTrue(np.allclose(U.todense(), U_known, rtol, atol))
self.assertTrue(np.allclose(rhs, rhs_known, rtol, atol))
self.assertTrue(np.allclose(theta, theta_known, rtol, atol))
def test_upwind_2d_full_beta_bc_dir(self):
f = np.array([[2, 2], [0, 2]])
gb = pp.meshing.cart_grid([f], [4, 2])
gb.assign_node_ordering()
gb.compute_geometry()
# define discretization
key = "transport"
variable = "T"
upwind = pp.Upwind(key)
upwind_coupling = pp.UpwindCoupling(key)
assign_discretization(gb, upwind, upwind_coupling, variable)
a = 1e-1
for g, d in gb:
aperture = np.ones(g.num_cells) * np.power(a, gb.dim_max() - g.dim)
specified_parameters = {"aperture": aperture}
bound_faces = g.tags["domain_boundary_faces"].nonzero()[0]
labels = np.array(["dir"] * bound_faces.size)
bc_val = np.zeros(g.num_faces)
bc_val[bound_faces] = 3
bound = pp.BoundaryCondition(g, bound_faces, labels)
specified_parameters.update({"bc": bound, "bc_values": bc_val})
pp.initialize_default_data(g, d, "transport", specified_parameters)
add_constant_darcy_flux(gb, upwind, [1, 1, 0], a)
assembler = pp.Assembler(gb)
assembler.discretize()
U_tmp, rhs = assembler.assemble_matrix_rhs()
grids = np.empty(gb.num_graph_nodes() + gb.num_graph_edges(),
dtype=np.object)
variables = np.empty_like(grids)
for g, d in gb:
grids[d["node_number"]] = g
variables[d["node_number"]] = variable
for e, d in gb.edges():
grids[d["edge_number"] + gb.num_graph_nodes()] = e
variables[d["edge_number"] + gb.num_graph_nodes()] = "lambda_u"
M, rhs = permute_matrix_vector(U_tmp, rhs, assembler.block_dof,
assembler.full_dof, grids, variables)
theta = np.linalg.solve(M.A, rhs)
M_known = np.array([
[
2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0
],
[
-1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 1.0
],
[
0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0,
0.0, 0.0
],
[
0.0, 0.0, -1.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0
],
[
-1.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0
],
[
0.0,
-1.0,
0.0,
0.0,
-1.0,
1.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
1.0,
0.0,
],
[
0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 1.0, 0.0,
0.0, 0.0
],
[
0.0,
0.0,
0.0,
-1.0,
0.0,
0.0,
-1.0,
2.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.1,
-0.1,
-1.0,
0.0,
-1.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.1,
0.0,
-1.0,
0.0,
-1.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
-1.0,
0.0,
0.0,
],
[
0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
-1.0, 0.0
],
[
0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, -1.0
],
])
rhs_known = np.array([
6.0, 3.0, 3.0, 3.0, 3.0, 0.0, 0.0, 0.0, 0.0, 0.3, 0.0, 0.0, 0.0,
0.0
])
theta_known = np.array([
3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, -3.0, -3.0, 3.0,
3.0
])
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(theta, theta_known, rtol, atol))
def test_upwind_2d_beta_positive(self):
f = np.array([[2, 2], [0, 2]])
gb = pp.meshing.cart_grid([f], [4, 2])
gb.assign_node_ordering()
gb.compute_geometry()
# define discretization
key = "transport"
upwind = pp.Upwind(key)
upwind_coupling = pp.UpwindCoupling(key)
variable = "T"
assign_discretization(gb, upwind, upwind_coupling, variable)
gb.add_node_props(["param"])
a = 1e-2
for g, d in gb:
aperture = np.ones(g.num_cells) * np.power(a, gb.dim_max() - g.dim)
bf = g.tags["domain_boundary_faces"].nonzero()[0]
bc = pp.BoundaryCondition(g, bf, bf.size * ["neu"])
specified_parameters = {"aperture": aperture, "bc": bc}
pp.initialize_default_data(g, d, "transport", specified_parameters)
add_constant_darcy_flux(gb, upwind, [2, 0, 0], a)
assembler = pp.Assembler(gb)
assembler.discretize()
U_tmp, rhs = assembler.assemble_matrix_rhs()
grids = np.empty(gb.num_graph_nodes() + gb.num_graph_edges(),
dtype=np.object)
variables = np.empty_like(grids)
for g, d in gb:
grids[d["node_number"]] = g
variables[d["node_number"]] = variable
for e, d in gb.edges():
grids[d["edge_number"] + gb.num_graph_nodes()] = e
variables[d["edge_number"] + gb.num_graph_nodes()] = "lambda_u"
M, rhs = permute_matrix_vector(U_tmp, rhs, assembler.block_dof,
assembler.full_dof, grids, variables)
# add generic mass matrix to solve system
I_diag = np.zeros(M.shape[0])
I_diag[:gb.num_cells()] = 1
I = np.diag(I_diag)
theta = np.linalg.solve(M + I, I_diag + rhs)
M_known = np.array([
[
2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0
],
[
-2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 1.0
],
[
0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0,
0.0, 0.0
],
[
0.0, 0.0, -2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0
],
[
0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0
],
[
0.0, 0.0, 0.0, 0.0, -2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
1.0, 0.0
],
[
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 1.0, 0.0,
0.0, 0.0
],
[
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -2.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
-1.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-1.0,
0.0,
-1.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-2.0,
0.0,
-1.0,
0.0,
0.0,
0.0,
],
[
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-2.0,
0.0,
-1.0,
0.0,
0.0,
],
[
0.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
-1.0, 0.0
],
[
0.0, 2.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, -1.0
],
])
rhs_known = np.zeros(14)
theta_known = np.array([
0.33333333,
0.55555556,
0.80246914,
2.60493827,
0.33333333,
0.55555556,
0.80246914,
2.60493827,
0.7037037,
0.7037037,
-1.40740741,
-1.40740741,
1.11111111,
1.11111111,
])
rtol = 1e-8
atol = rtol
self.assertTrue(np.allclose(M.A, M_known, rtol, atol))
self.assertTrue(np.allclose(rhs, rhs_known, rtol, atol))
self.assertTrue(np.allclose(theta, theta_known, rtol, atol))
def test_upwind_example_3(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
def funp_ex(pt):
return -np.sin(pt[0]) * np.sin(pt[1]) - pt[0]
g = pp.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
# Permeability
perm = pp.SecondOrderTensor(kxx=np.ones(g.num_cells))
# Boundaries
b_faces = g.get_all_boundary_faces()
bc = pp.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])
specified_parameters = {
"bc": bc,
"bc_values": bc_val,
"second_order_tensor": perm,
}
data = pp.initialize_default_data(g, {}, "flow", specified_parameters)
solver = pp.MVEM("flow")
solver.discretize(g, data)
D_flow, b_flow = solver.assemble_matrix_rhs(g, data)
solver_source = pp.DualScalarSource("flow")
solver_source.discretize(g, data)
D_source, b_source = solver_source.assemble_matrix_rhs(g, data)
up = sps.linalg.spsolve(D_flow + D_source, b_flow + b_source)
_, u = solver.extract_pressure(g, up, data), solver.extract_flux(g, up, data)
# Darcy_Flux
dis = u
# Boundaries
bc = pp.BoundaryCondition(g, b_faces, ["dir"] * b_faces.size)
bc_val = np.hstack(([1], np.zeros(g.num_faces - 1)))
specified_parameters = {"bc": bc, "bc_values": bc_val, "darcy_flux": dis}
data = pp.initialize_default_data(g, {}, "transport", specified_parameters)
# Advect solver
advect = pp.Upwind("transport")
advect.discretize(g, data)
U, rhs = advect.assemble_matrix_rhs(g, data)
time_step = advect.cfl(g, data)
rhs = time_step * rhs
U = time_step * U
data[pp.PARAMETERS]["transport"]["time_step"] = time_step
mass = pp.MassMatrix("transport")
mass.discretize(g, data)
M, _ = mass.assemble_matrix_rhs(g, data)
conc = np.zeros(g.num_cells)
M_minus_U = M - U
inv_mass = pp.InvMassMatrix("transport")
inv_mass.discretize(g, data)
invM, _ = inv_mass.assemble_matrix_rhs(g, data)
# Loop over the time
Nt = int(T / time_step)
time = np.empty(Nt)
for i in np.arange(Nt):
# Update the solution
conc = invM.dot((M_minus_U).dot(conc) + rhs)
time[i] = time_step * i
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)
def advdiff(gb, discr, param, bc_flag):
model = "transport"
model_data_adv, model_data_diff, model_data_src = data_advdiff(
gb, model, param, bc_flag
)
# discretization operator names
adv_id = "advection"
diff_id = "diffusion"
src_id = "source"
# variable names
variable = "scalar"
mortar_adv = "lambda_" + variable + "_" + adv_id
mortar_diff = "lambda_" + variable + "_" + diff_id
# save variable name for the post-process
param["scalar"] = variable
discr_adv = pp.Upwind(model_data_adv)
discr_adv_interface = pp.CellDofFaceDofMap(model_data_adv)
discr_diff = pp.Tpfa(model_data_diff)
discr_diff_interface = pp.CellDofFaceDofMap(model_data_diff)
coupling_adv = pp.UpwindCoupling(model_data_adv)
coupling_diff = pp.FluxPressureContinuity(
model_data_diff, discr_diff, discr_diff_interface
)
# mass term
mass_id = "mass"
discr_mass = pp.MassMatrix(model_data_adv)
discr_mass_interface = pp.CellDofFaceDofMap(model_data_adv)
discr_src = pp.ScalarSource(model_data_src)
for g, d in gb:
d[pp.PRIMARY_VARIABLES] = {variable: {"cells": 1}}
if g.dim == gb.dim_max():
d[pp.DISCRETIZATION] = {
variable: {adv_id: discr_adv,
diff_id: discr_diff,
mass_id: discr_mass,
src_id: discr_src}
}
else:
d[pp.DISCRETIZATION] = {
variable: {adv_id: discr_adv_interface,
diff_id: discr_diff_interface,
mass_id: discr_mass_interface}
}
for e, d in gb.edges():
g_slave, g_master = gb.nodes_of_edge(e)
d[pp.PRIMARY_VARIABLES] = {mortar_adv: {"cells": 1}, mortar_diff: {"cells": 1}}
d[pp.COUPLING_DISCRETIZATION] = {
adv_id: {
g_slave: (variable, adv_id),
g_master: (variable, adv_id),
e: (mortar_adv, coupling_adv),
},
diff_id: {
g_slave: (variable, diff_id),
g_master: (variable, diff_id),
e: (mortar_diff, coupling_diff),
},
}
# setup the advection-diffusion problem
assembler = pp.Assembler(gb, active_variables=[variable, mortar_diff, mortar_adv])
logger.info("Assemble the advective and diffusive terms of the transport problem")
block_A, block_b = assembler.assemble_matrix_rhs(add_matrices=False)
logger.info("done")
# unpack the matrices just computed
diff_name = diff_id + "_" + variable
adv_name = adv_id + "_" + variable
mass_name = mass_id + "_" + variable
source_name = src_id + "_" + variable
diff_coupling_name = diff_id + "_" + mortar_diff + "_" + variable + "_" + variable
adv_coupling_name = adv_id + "_" + mortar_adv + "_" + variable + "_" + variable
# need a sign for the convention of the conservation equation
M = block_A[mass_name]
A = block_A[diff_name] + block_A[diff_coupling_name] + \
block_A[adv_name] + block_A[adv_coupling_name]
b = block_b[diff_name] + block_b[diff_coupling_name] + \
block_b[adv_name] + block_b[adv_coupling_name] + \
block_b[source_name]
M_t = M.copy() / param["time_step"] * param.get("mass_weight", 1)
M_r = M.copy() * param.get("reaction", 0)
# Perform an LU factorization to speedup the solver
IE_solver = sps.linalg.factorized((M_t + A + M_r).tocsc())
# time loop
logger.info("Prepare the exporting")
save = pp.Exporter(gb, "solution", folder=param["folder"])
logger.info("done")
variables = [variable, param["pressure"], "frac_num", "cell_volumes"]
if discr["scheme"] is pp.MVEM or discr["scheme"] is pp.RT0:
variables.append(param["P0_flux"])
# assign the initial condition
x = np.zeros(A.shape[0])
assembler.distribute_variable(x)
for g, d in gb:
if g.dim == gb.dim_max():
d[pp.STATE][variable] = param.get("init_trans", 0) * np.ones(g.num_cells)
x = assembler.merge_variable(variable)
outflow = np.zeros(param["n_steps"])
logger.info("Start the time loop with " + str(param["n_steps"]) + " steps")
for i in np.arange(param["n_steps"]):
logger.info("Solve the linear system for time step " + str(i))
x = IE_solver(b + M_t.dot(x))
logger.info("done")
logger.info("Variable post-process")
assembler.distribute_variable(x)
logger.info("done")
logger.info("Export variable")
save.write_vtk(variables, time_step=i)
logger.info("done")
logger.info("Compute the production")
outflow[i] = compute_outflow(gb, param)
logger.info("done")
time = np.arange(param["n_steps"]) * param["time_step"]
save.write_pvd(time)
logger.info("Save outflow on file")
file_out = param["folder"] + "/outflow.csv"
data = np.vstack((time, outflow)).T
np.savetxt(file_out, data, delimiter=",")
logger.info("done")
logger.info("Save dof on file")
file_out = param["folder"] + "/dof_transport.csv"
np.savetxt(file_out, get_dof(assembler), delimiter=",", fmt="%d")
logger.info("done")