def test_linear_pressure_part_neumann_conditions(self):
g = self.grid()
bf = np.where(g.tags["domain_boundary_faces"].ravel())[0]
bc_type = bf.size * ["neu"]
# Set Dirichlet conditions. Note that indices are relative to bf,
# that is, counting only boundary faces.
bc_type[0] = "dir"
bc_type[2] = "dir" # Not [3]
bound = pp.BoundaryCondition(g, bf, bc_type)
fd = pp.Tpfa("flow")
bc_val = np.zeros(g.num_faces)
# Set up unit flow in x-direction, thus pressure gradient the other way
bc_val[[2, 5]] = 1
data = make_dictionary(g, bound, bc_val)
p = self.pressure(fd, g, data)
matrix_dictionary = data[pp.DISCRETIZATION_MATRICES]["flow"]
bound_p = (
matrix_dictionary["bound_pressure_cell"] * p
+ matrix_dictionary["bound_pressure_face"] * bc_val
)
self.assertTrue(np.allclose(bound_p[bf], -g.face_centers[0, bf]))
def _tpfa_matrix(g, perm=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
Returns
-------
out: sparse matrix
Two-point flux approximation matrix
"""
if isinstance(g, grid_bucket.GridBucket):
g = g.get_grids(lambda g_: g_.dim == g.dim_max())[0]
if perm is None:
perm = pp.SecondOrderTensor(np.ones(g.num_cells))
solver = pp.Tpfa("flow")
specified_parameters = {
"second_order_tensor": perm,
"bc": pp.BoundaryCondition(g, np.empty(0), ""),
}
data = pp.initialize_default_data(g, {}, "flow", specified_parameters)
solver.discretize(g, data)
return solver.assemble_matrix(g, data)
def __init__(self, gb, flow, model="flow"):
self.model = model
self.gb = gb
self.data = None
self.assembler = None
# discretization operator name
self.discr_name = self.model + "_flux"
self.discr = pp.Tpfa(self.model)
self.coupling_name = self.discr_name + "_coupling"
self.coupling = pp.RobinCoupling(self.model, self.discr)
self.source_name = self.model + "_source"
self.source = pp.ScalarSource(self.model)
# master variable name
self.variable = self.model + "_variable"
self.mortar = self.model + "_lambda"
# post process variables
self.pressure = "pressure"
self.flux = "darcy_flux" # it has to be this one
self.P0_flux = "P0_darcy_flux"
def test_linear_pressure_part_neumann_conditions_smaller_domain(self):
# Smaller domain, check that the smaller pressure gradient is captured
g = pp.CartGrid([2, 2], physdims=[1, 2])
g.compute_geometry()
bf = np.where(g.tags["domain_boundary_faces"].ravel())[0]
bc_type = bf.size * ["neu"]
bc_type[0] = "dir"
bc_type[2] = "dir"
bound = pp.BoundaryCondition(g, bf, bc_type)
fd = pp.Tpfa("flow")
bc_val = np.zeros(g.num_faces)
# Set up unit pressure gradient in x-direction
bc_val[[2, 5]] = 1
data = make_dictionary(g, bound, bc_val)
p = self.pressure(fd, g, data)
matrix_dictionary = data[pp.DISCRETIZATION_MATRICES]["flow"]
bound_p = (
matrix_dictionary["bound_pressure_cell"] * p
+ matrix_dictionary["bound_pressure_face"] * bc_val
)
self.assertTrue(np.allclose(bound_p[bf], -g.face_centers[0, bf]))
def solve(self, gb, method=None):
key = "flow"
if method is None:
discretization = pp.Tpfa(key)
elif method == "mpfa":
discretization = pp.Mpfa(key)
elif method == "mvem":
discretization = pp.MVEM(key)
assembler = test_utils.setup_flow_assembler(gb, discretization, key)
assembler.discretize()
A_flow, b_flow = assembler.assemble_matrix_rhs()
p = sps.linalg.spsolve(A_flow, b_flow)
assembler.distribute_variable(p)
if method == "mvem":
g2d = gb.grids_of_dimension(2)[0]
p_n = np.zeros(gb.num_cells())
for g, d in gb:
if g.dim == 2:
p_n[:g2d.num_cells] = d[pp.STATE]["pressure"][g.num_faces:]
else:
p_n[g2d.num_cells:] = d[pp.STATE]["pressure"][g.num_faces:]
d[pp.STATE]["pressure"] = d[pp.STATE]["pressure"][g.num_faces:]
p = p_n
return p
def test_symmetry_periodic_pressure_field_2d(method):
"""
Test that we obtain a symmetric solution accross the periodic boundary.
The test consider the unit square with periodic boundary conditions
on the top and bottom boundary. A source is added to the bottom row of
cells and we test that the solution is periodic.
Setup, with x denoting the source:
--------
| |
p = 0 | | p = 0
| x |
-------
"""
# Structured Cartesian grid
g, kxx = setup_cart_2d(np.array([5, 5]), [1, 1])
bot_faces = np.argwhere(g.face_centers[1] < 1e-5).ravel()
top_faces = np.argwhere(g.face_centers[1] > 1 - 1e-5).ravel()
left_faces = np.argwhere(g.face_centers[0] < 1e-5).ravel()
right_faces = np.argwhere(g.face_centers[0] > 1 - 1e-5).ravel()
dir_faces = np.hstack((left_faces, right_faces))
g.set_periodic_map(np.vstack((bot_faces, top_faces)))
bound = pp.BoundaryCondition(g, dir_faces, "dir")
# Solve
key = "flow"
d = pp.initialize_default_data(
g, {}, key, {
"second_order_tensor": pp.SecondOrderTensor(kxx),
"bc": bound
})
if method == "tpfa":
discr = pp.Tpfa(key)
elif method == "mpfa":
discr = pp.Mpfa(key)
else:
assert False
discr.discretize(g, d)
matrix_dictionary = d[pp.DISCRETIZATION_MATRICES][key]
flux, bound_flux = matrix_dictionary["flux"], matrix_dictionary[
"bound_flux"]
div = g.cell_faces.T
a = div * flux
pr_bound = np.zeros(g.num_faces)
src = np.zeros(g.num_cells)
src[2] = 1
rhs = -div * bound_flux * pr_bound + src
pr = np.linalg.solve(a.todense(), rhs)
p_diff = pr[5:15] - np.hstack((pr[-5:], pr[-10:-5]))
assert np.max(np.abs(p_diff)) < 1e-10
def setup_discr_tpfa(gb, key="flow"):
""" Setup the discretization Tpfa. """
discr = pp.Tpfa(key)
p_trace = pp.CellDofFaceDofMap(key)
interface = pp.FluxPressureContinuity(key, discr, p_trace)
for g, d in gb:
if g.dim == gb.dim_max():
d[pp.PRIMARY_VARIABLES] = {key: {"cells": 1}}
d[pp.DISCRETIZATION] = {key: {"flux": discr}}
else:
d[pp.PRIMARY_VARIABLES] = {key: {"cells": 1}}
d[pp.DISCRETIZATION] = {key: {"flux": p_trace}}
for e, d in gb.edges():
g_slave, g_master = gb.nodes_of_edge(e)
d[pp.PRIMARY_VARIABLES] = {key: {"cells": 1}}
d[pp.COUPLING_DISCRETIZATION] = {
"flux": {
g_slave: (key, "flux"),
g_master: (key, "flux"),
e: (key, interface),
}
}
return pp.Assembler(gb), (discr, p_trace)
def set_params_disrcetize(g, ambient_dim, method, periodic=False):
g.compute_geometry()
keyword = "flow"
if periodic:
south = g.face_centers[1] < np.min(g.nodes[1]) + 1e-8
north = g.face_centers[1] > np.max(g.nodes[1]) - 1e-8
bc = pp.BoundaryCondition(g, north + south, "per")
south_idx = np.argwhere(south).ravel()
north_idx = np.argwhere(north).ravel()
bc.set_periodic_map(np.vstack((south_idx, north_idx)))
else:
bc = pp.BoundaryCondition(g)
k = pp.SecondOrderTensor(np.ones(g.num_cells))
params = {
"bc": bc,
"second_order_tensor": k,
"mpfa_inverter": "python",
"ambient_dimension": ambient_dim,
}
data = pp.initialize_data(g, {}, keyword, params)
if method == "mpfa":
discr = pp.Mpfa(keyword)
elif method == "tpfa":
discr = pp.Tpfa(keyword)
discr.discretize(g, data)
flux = data[pp.DISCRETIZATION_MATRICES][keyword][discr.flux_matrix_key]
vector_source = data[pp.DISCRETIZATION_MATRICES][keyword][
discr.vector_source_matrix_key]
div = pp.fvutils.scalar_divergence(g)
return flux, vector_source, div
def test_two_cart_grids(self):
"""
We set up the test case -----|---------
| | |
| g1 | g2 |
| | |
-----|---------
with a linear pressure increase from left to right
"""
n = 2
xmax = 3
ymax = 1
split = 2
gb = self.generate_grids(n, xmax, ymax, split)
tol = 1e-6
for g, d in gb:
left = g.face_centers[0] < tol
right = g.face_centers[0] > xmax - tol
dir_bc = left + right
bound = pp.BoundaryCondition(g, dir_bc, "dir")
bc_val = np.zeros(g.num_faces)
bc_val[left] = xmax
bc_val[right] = 0
specified_parameters = {"bc": bound, "bc_values": bc_val}
pp.initialize_default_data(g, d, "flow", specified_parameters)
for e, d in gb.edges():
mg = d["mortar_grid"]
d[pp.PARAMETERS] = pp.Parameters(mg, ["flow"], [{}])
pp.params.data.add_discretization_matrix_keyword(d, "flow")
# assign discretization
data_key = "flow"
tpfa = pp.Tpfa(data_key)
coupler = pp.FluxPressureContinuity(data_key, tpfa)
assembler = test_utils.setup_flow_assembler(gb,
tpfa,
data_key,
coupler=coupler)
test_utils.solve_and_distribute_pressure(gb, assembler)
# test pressure
for g, d in gb:
self.assertTrue(
np.allclose(d[pp.STATE]["pressure"], xmax - g.cell_centers[0]))
# test mortar solution
for e, d_e in gb.edges():
mg = d_e["mortar_grid"]
g2, g1 = gb.nodes_of_edge(e)
master_to_m = mg.master_to_mortar_avg()
slave_to_m = mg.slave_to_mortar_avg()
master_area = master_to_m * g1.face_areas
slave_area = slave_to_m * g2.face_areas
self.assertTrue(
np.allclose(d_e[pp.STATE]["mortar_flux"] / master_area, 1))
self.assertTrue(
np.allclose(d_e[pp.STATE]["mortar_flux"] / slave_area, 1))
def homo_tpfa(g):
return {
"scheme": pp.Tpfa("flow"),
"dof": {
"cells": 1
},
"label": "homo_tpfa"
}
def test_uniform_flow_cart_2d_1d_simplex(self):
# Unstructured simplex grid
gb = setup_2d_1d(np.array([10, 10]), simplex_grid=True)
key = "flow"
tpfa = pp.Tpfa(key)
assembler = test_utils.setup_flow_assembler(gb, tpfa, key)
test_utils.solve_and_distribute_pressure(gb, assembler)
self.assertTrue(check_pressures(gb))
def solve_tpfa(gb, folder, discr_3d=None):
# Choose and define the solvers and coupler
flow_discretization = pp.Tpfa("flow")
source_discretization = pp.ScalarSource("flow")
run_flow(gb,
flow_discretization,
source_discretization,
folder,
is_FV=True,
discr_3d=discr_3d)
def test_fv_cart_2d_periodic(method):
""" Apply TPFA and MPFA on a periodic Cartesian grid, should obtain Laplacian stencil."""
# Set up 3 X 3 Cartesian grid
nx = np.array([3, 3])
g = pp.CartGrid(nx)
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = pp.SecondOrderTensor(kxx)
left_faces = [0, 4, 8, 12, 13, 14]
right_faces = [3, 7, 11, 21, 22, 23]
periodic_face_map = np.vstack((left_faces, right_faces))
g.set_periodic_map(periodic_face_map)
bound = pp.BoundaryCondition(g)
key = "flow"
d = pp.initialize_default_data(g, {}, key, {
"second_order_tensor": perm,
"bc": bound
})
if method == "tpfa":
discr = pp.Tpfa(key)
elif method == "mpfa":
discr = pp.Mpfa(key)
else:
assert False
discr.discretize(g, d)
matrix_dictionary = d[pp.DISCRETIZATION_MATRICES][key]
trm, bound_flux = matrix_dictionary["flux"], matrix_dictionary[
"bound_flux"]
div = g.cell_faces.T
a = div * trm
b = -(div * bound_flux).A
# Create laplace matrix
A_lap = np.array([
[4.0, -1.0, -1.0, -1.0, 0.0, 0.0, -1.0, 0.0, 0.0],
[-1.0, 4.0, -1.0, 0.0, -1.0, 0.0, 0.0, -1.0, 0.0],
[-1.0, -1.0, 4.0, 0.0, 0.0, -1.0, 0.0, 0.0, -1.0],
[-1.0, 0.0, 0.0, 4.0, -1.0, -1.0, -1.0, 0.0, 0.0],
[0.0, -1.0, 0.0, -1.0, 4.0, -1.0, 0.0, -1.0, 0.0],
[0.0, 0.0, -1.0, -1.0, -1.0, 4.0, 0.0, 0.0, -1.0],
[-1.0, 0.0, 0.0, -1.0, 0.0, 0.0, 4.0, -1.0, -1.0],
[0.0, -1.0, 0.0, 0.0, -1.0, 0.0, -1.0, 4.0, -1.0],
[0.0, 0.0, -1.0, 0.0, 0.0, -1.0, -1.0, -1.0, 4.0],
])
assert np.allclose(a.A, A_lap)
assert np.allclose(b, 0)
return a
def set_param_flow(self, gb, no_flow=False, kn=1e3, method="mpfa"):
# Set up flow field with uniform flow in y-direction
kw = "flow"
for g, d in gb:
parameter_dictionary = {}
perm = pp.SecondOrderTensor(kxx=np.ones(g.num_cells))
parameter_dictionary["second_order_tensor"] = perm
b_val = np.zeros(g.num_faces)
if g.dim == 2:
bound_faces = pp.face_on_side(g, ["ymin", "ymax"])
if no_flow:
b_val[bound_faces[0]] = 1
b_val[bound_faces[1]] = 1
bound_faces = np.hstack((bound_faces[0], bound_faces[1]))
labels = np.array(["dir"] * bound_faces.size)
parameter_dictionary["bc"] = pp.BoundaryCondition(
g, bound_faces, labels)
y_max_faces = pp.face_on_side(g, "ymax")[0]
b_val[y_max_faces] = 1
else:
parameter_dictionary["bc"] = pp.BoundaryCondition(g)
parameter_dictionary["bc_values"] = b_val
parameter_dictionary["mpfa_inverter"] = "python"
d[pp.PARAMETERS] = pp.Parameters(g, [kw], [parameter_dictionary])
d[pp.DISCRETIZATION_MATRICES] = {"flow": {}}
gb.add_edge_props("kn")
for e, d in gb.edges():
mg = d["mortar_grid"]
flow_dictionary = {
"normal_diffusivity": 2 * kn * np.ones(mg.num_cells)
}
d[pp.PARAMETERS] = pp.Parameters(keywords=["flow"],
dictionaries=[flow_dictionary])
d[pp.DISCRETIZATION_MATRICES] = {"flow": {}}
discretization_key = kw + "_" + pp.DISCRETIZATION
for g, d in gb:
# Choose discretization and define the solver
if method == "mpfa":
discr = pp.Mpfa(kw)
elif method == "mvem":
discr = pp.MVEM(kw)
else:
discr = pp.Tpfa(kw)
d[discretization_key] = discr
for _, d in gb.edges():
d[discretization_key] = pp.RobinCoupling(kw, discr)
def elliptic_disc(gb, keyword, use_mpfa=False):
"""
Discretize the elliptic operator on each graph node
"""
for g, d in gb:
if use_mpfa:
pp.Mpfa(keyword).discretize(g, d)
else:
pp.Tpfa(keyword).discretize(g, d)
d[pp.DISCRETIZATION_MATRICES][keyword][
"div"] = pp.fvutils.scalar_divergence(g)
def test_periodic_pressure_field_2d(method):
"""
Test that TPFA approximate an analytical periodic solution by imposing
periodic boundary conditions to the bottom and top faces of the unit square.
"""
# Structured Cartesian grid
g, kxx = setup_cart_2d(np.array([10, 10]), [1, 1])
bot_faces = np.argwhere(g.face_centers[1] < 1e-5).ravel()
top_faces = np.argwhere(g.face_centers[1] > 1 - 1e-5).ravel()
left_faces = np.argwhere(g.face_centers[0] < 1e-5).ravel()
right_faces = np.argwhere(g.face_centers[0] > 1 - 1e-5).ravel()
dir_faces = np.hstack((left_faces, right_faces))
g.set_periodic_map(np.vstack((bot_faces, top_faces)))
bound = pp.BoundaryCondition(g, dir_faces, "dir")
# Solve
key = "flow"
d = pp.initialize_default_data(
g, {}, key, {
"second_order_tensor": pp.SecondOrderTensor(kxx),
"bc": bound
})
if method == "tpfa":
discr = pp.Tpfa(key)
elif method == "mpfa":
discr = pp.Mpfa(key)
else:
assert False
discr.discretize(g, d)
matrix_dictionary = d[pp.DISCRETIZATION_MATRICES][key]
flux, bound_flux = matrix_dictionary["flux"], matrix_dictionary[
"bound_flux"]
div = g.cell_faces.T
a = div * flux
pr_bound, pr_cell, src = setup_periodic_pressure_field(g, kxx)
rhs = -div * bound_flux * pr_bound + src * g.cell_volumes
pr = np.linalg.solve(a.todense(), rhs)
p_diff = pr - pr_cell
assert np.max(np.abs(p_diff)) < 0.06
def solve(self, method):
key = "flow"
gb = self.gb
if method == "tpfa":
discretization = pp.Tpfa(key)
elif method == "mpfa":
discretization = pp.Mpfa(key)
elif method == "mvem":
discretization = pp.MVEM(key)
assembler = test_utils.setup_flow_assembler(gb, discretization, key)
assembler.discretize()
A_flow, b_flow = assembler.assemble_matrix_rhs()
p = sps.linalg.spsolve(A_flow, b_flow)
assembler.distribute_variable(p)
return p
def test_zero_pressure(self):
g = self.grid()
bf = np.where(g.tags["domain_boundary_faces"].ravel())[0]
bc_type = bf.size * ["dir"]
bound = pp.BoundaryCondition(g, bf, bc_type)
fd = pp.Tpfa("flow")
data = make_dictionary(g, bound)
p = self.pressure(fd, g, data)
self.assertTrue(np.allclose(p, np.zeros_like(p)))
matrix_dictionary = data[pp.DISCRETIZATION_MATRICES]["flow"]
bound_p = matrix_dictionary["bound_pressure_cell"] * p + matrix_dictionary[
"bound_pressure_face"
] * np.zeros(g.num_faces)
self.assertTrue(np.allclose(bound_p, np.zeros_like(bound_p)))
def diffusive_disc(self):
"Discretization of term \nabla K \nabla T"
class DiffusiveMixedDim(pp.TpfaMixedDim):
def __init__(self, physics="flow", has_advective_term=False):
pp.TpfaMixedDim.__init__(self, physics=physics)
if has_advective_term:
self.coupling_conditions = [None, self.coupling_conditions]
self.solver = pp.numerics.mixed_dim.coupler.Coupler(
self.discr, self.coupling_conditions)
if self.is_GridBucket:
diffusive_discr = DiffusiveMixedDim(self.physics,
self.advective_term)
else:
diffusive_discr = pp.Tpfa(physics=self.physics)
return diffusive_discr
def hete2(g):
if g.dim == 2:
scheme = {
"scheme": pp.MVEM("flow"),
"dof": {
"cells": 1,
"faces": 1
},
"label": "hete2"
}
else:
scheme = {
"scheme": pp.Tpfa("flow"),
"dof": {
"cells": 1
},
"label": "hete2"
}
return scheme
def test_linear_pressure_dirichlet_conditions(self):
g = self.grid()
bf = np.where(g.tags["domain_boundary_faces"].ravel())[0]
bc_type = bf.size * ["dir"]
bound = pp.BoundaryCondition(g, bf, bc_type)
fd = pp.Tpfa("flow")
bc_val = 1 * g.face_centers[0] + 2 * g.face_centers[1]
data = make_dictionary(g, bound, bc_val)
p = self.pressure(fd, g, data)
matrix_dictionary = data[pp.DISCRETIZATION_MATRICES]["flow"]
bound_p = (
matrix_dictionary["bound_pressure_cell"] * p
+ matrix_dictionary["bound_pressure_face"] * bc_val
)
self.assertTrue(np.allclose(bound_p[bf], bc_val[bf]))
def test_sign_trouble_two_neumann_sides(self):
g = pp.CartGrid(np.array([2, 2]), physdims=[2, 2])
g.compute_geometry()
bc_val = np.zeros(g.num_faces)
bc_val[[0, 3]] = 1
bc_val[[2, 5]] = -1
t = pp.Tpfa("flow")
data = make_dictionary(g, pp.BoundaryCondition(g), bc_val)
t.discretize(g, data)
t.assemble_matrix_rhs(g, data)
# The problem is singular, and spsolve does not work well on all systems.
# Instead, set a consistent solution, and check that the boundary
# pressure is recovered.
x = g.cell_centers[0]
matrix_dictionary = data[pp.DISCRETIZATION_MATRICES]["flow"]
bound_p = (
matrix_dictionary["bound_pressure_cell"] * x
+ matrix_dictionary["bound_pressure_face"] * bc_val
)
self.assertTrue(bound_p[0] == x[0] - 0.5)
self.assertTrue(bound_p[2] == x[1] + 0.5)
def test_constant_pressure_simplex_grid(self):
g = self.simplex_grid()
bf = np.where(g.tags["domain_boundary_faces"].ravel())[0]
bc_type = bf.size * ["dir"]
bound = pp.BoundaryCondition(g, bf, bc_type)
fd = pp.Tpfa("flow")
bc_val = np.ones(g.num_faces)
data = make_dictionary(g, bound, bc_val)
p = self.pressure(fd, g, data)
self.assertTrue(np.allclose(p, np.ones_like(p)))
matrix_dictionary = data[pp.DISCRETIZATION_MATRICES]["flow"]
bound_p = (
matrix_dictionary["bound_pressure_cell"] * p
+ matrix_dictionary["bound_pressure_face"] * bc_val
)
self.assertTrue(np.allclose(bound_p[bf], bc_val[bf]))
def test_linear_pressure_part_neumann_conditions_reverse_sign(self):
g = self.grid()
bf = np.where(g.tags["domain_boundary_faces"].ravel())[0]
bc_type = bf.size * ["neu"]
bc_type[0] = "dir"
bc_type[2] = "dir"
bound = pp.BoundaryCondition(g, bf, bc_type)
fd = pp.Tpfa("flow")
bc_val = np.zeros(g.num_faces)
# Set up pressure gradient in x-direction, with value -1
bc_val[[2, 5]] = -1
data = make_dictionary(g, bound, bc_val)
p = self.pressure(fd, g, data)
matrix_dictionary = data[pp.DISCRETIZATION_MATRICES]["flow"]
bound_p = (
matrix_dictionary["bound_pressure_cell"] * p
+ matrix_dictionary["bound_pressure_face"] * bc_val
)
self.assertTrue(np.allclose(bound_p[bf], g.face_centers[0, bf]))
def solve(self, kf, description, multi_point):
gb, domain = pp.grid_buckets_2d.benchmark_regular(
{"mesh_size_frac": 0.045})
# Assign parameters
setup.add_data(gb, domain, kf)
key = "flow"
if multi_point:
method = pp.Mpfa(key)
else:
method = pp.Tpfa(key)
coupler = pp.RobinCoupling(key, method)
for g, d in gb:
d[pp.PRIMARY_VARIABLES] = {"pressure": {"cells": 1}}
d[pp.DISCRETIZATION] = {"pressure": {"diffusive": method}}
for e, d in gb.edges():
g1, g2 = gb.nodes_of_edge(e)
d[pp.PRIMARY_VARIABLES] = {"mortar_solution": {"cells": 1}}
d[pp.COUPLING_DISCRETIZATION] = {
"lambda": {
g1: ("pressure", "diffusive"),
g2: ("pressure", "diffusive"),
e: ("mortar_solution", coupler),
}
}
d[pp.DISCRETIZATION_MATRICES] = {"flow": {}}
assembler = pp.Assembler(gb)
# Discretize
assembler.discretize()
A, b = assembler.assemble_matrix_rhs()
p = sps.linalg.spsolve(A, b)
assembler.distribute_variable(p)
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 solve(self, gb, analytic_p):
# Parameter-discretization keyword:
kw = "flow"
# Terms
key_flux = "flux"
key_src = "src"
# Primary variables
key_p = "pressure" # pressure name
key_m = "mortar" # mortar name
tpfa = pp.Tpfa(kw)
src = pp.ScalarSource(kw)
for g, d in gb:
d[pp.DISCRETIZATION] = {key_p: {key_src: src, key_flux: tpfa}}
d[pp.PRIMARY_VARIABLES] = {key_p: {"cells": 1}}
for e, d in gb.edges():
g1, g2 = gb.nodes_of_edge(e)
d[pp.PRIMARY_VARIABLES] = {key_m: {"cells": 1}}
if g1.dim == g2.dim:
mortar_disc = pp.FluxPressureContinuity(kw, tpfa)
else:
mortar_disc = pp.RobinCoupling(kw, tpfa)
d[pp.COUPLING_DISCRETIZATION] = {
key_flux: {
g1: (key_p, key_flux),
g2: (key_p, key_flux),
e: (key_m, mortar_disc),
}
}
assembler = pp.Assembler(gb)
assembler.discretize()
A, b = assembler.assemble_matrix_rhs()
x = sps.linalg.spsolve(A, b)
assembler.distribute_variable(x)
# test pressure
for g, d in gb:
ap, _, _ = analytic_p(g.cell_centers)
self.assertTrue(np.max(np.abs(d[pp.STATE][key_p] - ap)) < 5e-2)
# test mortar solution
for e, d_e in gb.edges():
mg = d_e["mortar_grid"]
g1, g2 = gb.nodes_of_edge(e)
if g1 == g2:
left_to_m = mg.master_to_mortar_avg()
right_to_m = mg.slave_to_mortar_avg()
else:
continue
d1 = gb.node_props(g1)
d2 = gb.node_props(g2)
_, analytic_flux, _ = analytic_p(g1.face_centers)
# the aperture is assumed constant
left_flux = np.sum(analytic_flux * g1.face_normals[:2], 0)
left_flux = left_to_m * (
d1[pp.DISCRETIZATION_MATRICES][kw]["bound_flux"] * left_flux
)
# right flux is negative lambda
right_flux = np.sum(analytic_flux * g2.face_normals[:2], 0)
right_flux = -right_to_m * (
d2[pp.DISCRETIZATION_MATRICES][kw]["bound_flux"] * right_flux
)
self.assertTrue(np.max(np.abs(d_e[pp.STATE][key_m] - left_flux)) < 5e-2)
self.assertTrue(np.max(np.abs(d_e[pp.STATE][key_m] - right_flux)) < 5e-2)
def set_parameters(
self,
neu_val_top=None,
dir_val_top=None,
kn: float = 1e0,
method="mpfa",
aperture: float = 1e-1,
gravity_angle: float = 0,
) -> None:
"""
Parameters:
neu_val_top (float): Default None implies Dirichlet on top. If not None,
the prescribed value will be applied as the Neumann bc value.
dir_val_top (float): If not None, the prescribed value will be
applied as the Dirichlet bc value. Note that neu_val_top takes precedent
over dir_val_top.
method: Discretization method.
kn: Normal permeability of the fracture. Will be multiplied by aperture/2
to yield the normal diffusivity.
aperture: Fracture aperture.
gravity_angle: Angle by which to rotate the applied vector source field.
"""
# Set up flow field with uniform flow in y-direction
kw = "flow"
gb = self.gb
for g, d in gb:
a = np.power(aperture, gb.dim_max() - g.dim)
perm = pp.SecondOrderTensor(kxx=a * np.ones(g.num_cells))
gravity = np.zeros((gb.dim_max(), g.num_cells))
# Angle of zero means force vector of [0, -1]
gravity[1, :] = -np.cos(gravity_angle)
gravity[0, :] = np.sin(gravity_angle)
b_val = np.zeros(g.num_faces)
if g.dim == self.gb.dim_max():
if neu_val_top is not None:
dir_faces = np.atleast_1d(pp.face_on_side(g, ["ymin"])[0])
neu_faces = np.atleast_1d(pp.face_on_side(g, ["ymax"])[0])
b_val[neu_faces] = neu_val_top
else:
dir_faces = pp.face_on_side(g, ["ymin", "ymax"])
b_val[dir_faces[0]] = 0
if dir_val_top is not None:
b_val[dir_faces[1]] = dir_val_top
dir_faces = np.hstack((dir_faces[0], dir_faces[1]))
labels = np.array(["dir"] * dir_faces.size)
bc = pp.BoundaryCondition(g, dir_faces, labels)
else:
bc = pp.BoundaryCondition(g)
parameter_dictionary = {
"bc_values": b_val,
"bc": bc,
"ambient_dimension": gb.dim_max(),
"mpfa_inverter": "python",
"second_order_tensor": perm,
"vector_source": gravity.ravel("F"),
}
pp.initialize_data(g, d, "flow", parameter_dictionary)
for e, d in gb.edges():
g1, g2 = gb.nodes_of_edge(e)
mg = d["mortar_grid"]
a = aperture * np.ones(mg.num_cells)
gravity = np.zeros((gb.dim_max(), mg.num_cells))
# Angle of zero means force vector of [0, -1]
gravity[1, :] = -np.cos(gravity_angle)
gravity[0, :] = np.sin(gravity_angle)
gravity *= a / 2
parameter_dictionary = {
"normal_diffusivity": 2 / a * kn,
"ambient_dimension": gb.dim_max(),
"vector_source": gravity.ravel("F"),
}
pp.initialize_data(mg, d, "flow", parameter_dictionary)
# discretization_key = kw + "_" + pp.DISCRETIZATION
for g, d in gb:
# Choose discretization and define the solver
if method == "mpfa":
discr = pp.Mpfa(kw)
elif method == "tpfa":
discr = pp.Tpfa(kw)
else:
raise ValueError("Unexpected discretization method ")
def test_md_flow():
# Three fractures, will create intersection lines and point
frac_1 = np.array([[2, 4, 4, 2], [3, 3, 3, 3], [0, 0, 6, 6]])
frac_2 = np.array([[3, 3, 3, 3], [2, 4, 4, 2], [0, 0, 6, 6]])
frac_3 = np.array([[0, 6, 6, 0], [2, 2, 4, 4], [3, 3, 3, 3]])
gb = pp.meshing.cart_grid(fracs=[frac_1, frac_2, frac_3],
nx=np.array([6, 6, 6]))
gb.compute_geometry()
pressure_variable = "pressure"
flux_variable = "mortar_flux"
keyword = "flow"
discr = pp.Tpfa(keyword)
source_discr = pp.ScalarSource(keyword)
coupling_discr = pp.RobinCoupling(keyword, discr, discr)
for g, d in gb:
# Assign data
if g.dim == gb.dim_max():
upper_left_ind = np.argmax(np.linalg.norm(g.face_centers, axis=0))
bc = pp.BoundaryCondition(g, np.array([0, upper_left_ind]),
["dir", "dir"])
bc_values = np.zeros(g.num_faces)
bc_values[0] = 1
sources = np.random.rand(g.num_cells) * g.cell_volumes
specified_parameters = {
"bc": bc,
"bc_values": bc_values,
"source": sources
}
else:
sources = np.random.rand(g.num_cells) * g.cell_volumes
specified_parameters = {"source": sources}
# Initialize data
pp.initialize_default_data(g, d, keyword, specified_parameters)
# Declare grid primary variable
d[pp.PRIMARY_VARIABLES] = {pressure_variable: {"cells": 1}}
# Assign discretization
d[pp.DISCRETIZATION] = {
pressure_variable: {
"diff": discr,
"source": source_discr
}
}
# Initialize state
d[pp.STATE] = {
pressure_variable: np.zeros(g.num_cells),
pp.ITERATE: {
pressure_variable: np.zeros(g.num_cells)
},
}
for e, d in gb.edges():
mg = d["mortar_grid"]
pp.initialize_data(mg, d, keyword, {"normal_diffusivity": 1})
d[pp.PRIMARY_VARIABLES] = {flux_variable: {"cells": 1}}
d[pp.COUPLING_DISCRETIZATION] = {}
d[pp.COUPLING_DISCRETIZATION]["coupling"] = {
e[0]: (pressure_variable, "diff"),
e[1]: (pressure_variable, "diff"),
e: (flux_variable, coupling_discr),
}
d[pp.STATE] = {
flux_variable: np.zeros(mg.num_cells),
pp.ITERATE: {
flux_variable: np.zeros(mg.num_cells)
},
}
dof_manager = pp.DofManager(gb)
assembler = pp.Assembler(gb, dof_manager)
assembler.discretize()
# Reference discretization
A_ref, b_ref = assembler.assemble_matrix_rhs()
manager = pp.ad.EquationManager(gb, dof_manager)
grid_list = [g for g, _ in gb]
edge_list = [e for e, _ in gb.edges()]
node_discr = pp.ad.MpfaAd(keyword, grid_list)
edge_discr = pp.ad.RobinCouplingAd(keyword, edge_list)
bc_val = pp.ad.BoundaryCondition(keyword, grid_list)
source = pp.ad.ParameterArray(param_keyword=keyword,
array_keyword='source',
grids=grid_list)
projections = pp.ad.MortarProjections(gb=gb)
div = pp.ad.Divergence(grids=grid_list)
p = manager.merge_variables([(g, pressure_variable) for g in grid_list])
lmbda = manager.merge_variables([(e, flux_variable) for e in edge_list])
flux = (node_discr.flux * p + node_discr.bound_flux * bc_val +
node_discr.bound_flux * projections.mortar_to_primary_int * lmbda)
flow_eq = div * flux - projections.mortar_to_secondary_int * lmbda - source
interface_flux = edge_discr.mortar_scaling * (
projections.primary_to_mortar_avg * node_discr.bound_pressure_cell * p
+ projections.primary_to_mortar_avg * node_discr.bound_pressure_face *
projections.mortar_to_primary_int * lmbda -
projections.secondary_to_mortar_avg * p +
edge_discr.mortar_discr * lmbda)
flow_eq_ad = pp.ad.Expression(flow_eq, dof_manager, "flow on nodes")
flow_eq_ad.discretize(gb)
interface_eq_ad = pp.ad.Expression(interface_flux, dof_manager,
"flow on interface")
manager.equations += [flow_eq_ad, interface_eq_ad]
state = np.zeros(gb.num_cells() + gb.num_mortar_cells())
A, b = manager.assemble_matrix_rhs(state=state)
diff = A - A_ref
if diff.data.size > 0:
assert np.max(np.abs(diff.data)) < 1e-10
assert np.max(np.abs(b - b_ref)) < 1e-10
def test_tpfa_cart_2d(self):
""" Apply TPFA on Cartesian grid, should obtain Laplacian stencil. """
# Set up 3 X 3 Cartesian grid
nx = np.array([3, 3])
g = pp.CartGrid(nx)
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = pp.SecondOrderTensor(kxx)
bound_faces = np.array([0, 3, 12])
bound = pp.BoundaryCondition(g, bound_faces,
["dir"] * bound_faces.size)
key = "flow"
d = pp.initialize_default_data(g, {}, key, {
"second_order_tensor": perm,
"bc": bound
})
discr = pp.Tpfa(key)
discr.discretize(g, d)
matrix_dictionary = d[pp.DISCRETIZATION_MATRICES][key]
trm, bound_flux = matrix_dictionary["flux"], matrix_dictionary[
"bound_flux"]
div = g.cell_faces.T
a = div * trm
b = -(div * bound_flux).A
# Checks on interior cell
mid = 4
self.assertTrue(a[mid, mid] == 4)
self.assertTrue(a[mid - 1, mid] == -1)
self.assertTrue(a[mid + 1, mid] == -1)
self.assertTrue(a[mid - 3, mid] == -1)
self.assertTrue(a[mid + 3, mid] == -1)
self.assertTrue(np.all(b[mid, :] == 0))
# The first cell should have two Dirichlet bnds
self.assertTrue(a[0, 0] == 6)
self.assertTrue(a[0, 1] == -1)
self.assertTrue(a[0, 3] == -1)
self.assertTrue(b[0, 0] == 2)
self.assertTrue(b[0, 12] == 2)
# Cell 3 has one Dirichlet, one Neumann face
self.assertTrue(a[2, 2] == 4)
self.assertTrue(a[2, 1] == -1)
self.assertTrue(a[2, 5] == -1)
self.assertTrue(b[2, 3] == 2)
self.assertTrue(b[2, 14] == -1)
# Cell 2 has one Neumann face
self.assertTrue(a[1, 1] == 3)
self.assertTrue(a[1, 0] == -1)
self.assertTrue(a[1, 2] == -1)
self.assertTrue(a[1, 4] == -1)
self.assertTrue(b[1, 13] == -1)
return a