def coarsening_example0(**kwargs):
#######################
# Simple 2d coarsening based on tpfa for Cartesian grids
# isotropic permeability
#######################
Nx = Ny = 7
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
if kwargs["visualize"]:
plot_grid(g, info="all", alpha=0)
part = create_partition(tpfa_matrix(g))
g = generate_coarse_grid(g, part)
g.compute_geometry(is_starshaped=True)
if kwargs["visualize"]:
plot_grid(g, info="all", alpha=0)
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
part = create_partition(tpfa_matrix(g), cdepth=3)
g = generate_coarse_grid(g, part)
g.compute_geometry(is_starshaped=True)
if kwargs["visualize"]:
plot_grid(g, info="all", alpha=0)
def setup_2d():
grid_list = []
# Unstructured perturbation
nx = np.array([2, 2])
g_cart_unpert = structured.CartGrid(nx)
g_cart_unpert.compute_geometry()
grid_list.append(g_cart_unpert)
# Structured perturbation
g_cart_spert = structured.CartGrid(nx)
g_cart_spert.nodes[0, 4] = 1.5
g_cart_spert.compute_geometry()
grid_list.append(g_cart_spert)
# Larger grid, random perturbations
nx = np.array([3, 3])
g_cart_rpert = structured.CartGrid(nx)
dx = 1
pert = .4
rand = np.vstack(
(
np.random.rand(g_cart_rpert.dim, g_cart_rpert.num_nodes),
np.repeat(0., g_cart_rpert.num_nodes),
)
)
g_cart_rpert.nodes = g_cart_rpert.nodes + dx * pert * (0.5 - rand)
g_cart_rpert.compute_geometry()
grid_list.append(g_cart_rpert)
return grid_list
def setup_2d():
grid_list = []
# Unstructured perturbation
nx = np.array([2, 2])
g_cart_unpert = structured.CartGrid(nx)
g_cart_unpert.compute_geometry()
grid_list.append(g_cart_unpert)
# Structured perturbation
g_cart_spert = structured.CartGrid(nx)
g_cart_spert.nodes[0, 4] = 1.5
g_cart_spert.compute_geometry()
grid_list.append(g_cart_spert)
# Larger grid, random perturbations
nx = np.array([3, 3])
g_cart_rpert = structured.CartGrid(nx)
dx = 1
pert = 0.4
rand = np.vstack(
(
np.random.rand(g_cart_rpert.dim, g_cart_rpert.num_nodes),
np.repeat(0.0, g_cart_rpert.num_nodes),
)
)
g_cart_rpert.nodes = g_cart_rpert.nodes + dx * pert * (0.5 - rand)
# No perturbations of the z-coordinate (which is not active in this case)
g_cart_rpert.nodes[2, :] = 0
g_cart_rpert.compute_geometry()
grid_list.append(g_cart_rpert)
return grid_list
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_uniform_displacement_neumann(self):
physdims = [1, 1]
g_size = [4, 8]
g_list = [structured.CartGrid([n, n], physdims=physdims) for n in g_size]
[g.compute_geometry() for g in g_list]
error = []
for g in g_list:
bot = np.ravel(np.argwhere(g.face_centers[1, :] < 1e-10))
left = np.ravel(np.argwhere(g.face_centers[0, :] < 1e-10))
dir_faces = np.hstack((left, bot))
bound = bc.BoundaryCondition(
g, dir_faces.ravel("F"), ["dir"] * dir_faces.size
)
constit = setup_stiffness(g)
# Python inverter is most efficient for small problems
stress, bound_stress = mpsa.mpsa(g, constit, bound, inverter="python")
div = fvutils.vector_divergence(g)
a = div * stress
d_x = np.random.rand(1)
d_y = np.random.rand(1)
d_bound = np.zeros((g.dim, g.num_faces))
d_bound[0, bound.is_dir] = d_x
d_bound[1, bound.is_dir] = d_y
rhs = div * bound_stress * d_bound.ravel("F")
d = np.linalg.solve(a.todense(), -rhs)
traction = stress * d + bound_stress * d_bound.ravel("F")
self.assertTrue(np.max(np.abs(d[::2] - d_x)) < 1e-8)
self.assertTrue(np.max(np.abs(d[1::2] - d_y)) < 1e-8)
self.assertTrue(np.max(np.abs(traction)) < 1e-8)
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_coarse_grid_2d(self):
g = structured.CartGrid([3, 2])
g.compute_geometry()
co.generate_coarse_grid(g, [5, 2, 2, 5, 2, 2])
assert g.num_cells == 2
assert g.num_faces == 12
assert g.num_nodes == 11
pt = np.tile(np.array([2, 1, 0]), (g.nodes.shape[1], 1)).T
find = np.isclose(pt, g.nodes).all(axis=0)
assert find.any() == False
faces_cell0, _, orient_cell0 = sps.find(g.cell_faces[:, 0])
assert np.array_equal(faces_cell0, [1, 2, 4, 5, 7, 8, 10, 11])
assert np.array_equal(orient_cell0, [-1, 1, -1, 1, -1, -1, 1, 1])
faces_cell1, _, orient_cell1 = sps.find(g.cell_faces[:, 1])
assert np.array_equal(faces_cell1, [0, 1, 3, 4, 6, 9])
assert np.array_equal(orient_cell1, [-1, 1, -1, 1, -1, 1])
known = np.array([[0, 4], [1, 5], [3, 6], [4, 7], [5, 8], [6, 10],
[0, 1], [1, 2], [2, 3], [7, 8], [8, 9], [9, 10]])
for f in np.arange(g.num_faces):
assert np.array_equal(sps.find(g.face_nodes[:, f])[0], known[f, :])
def test_cart_3d(self):
g = structured.CartGrid([4, 3, 3])
g.nodes = g.nodes + 0.2 * np.random.random((g.dim, g.nodes.shape[1]))
g.compute_geometry()
# Pick out cells (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 2), (1, 0, 2)
c = np.array([0, 1, 4, 12, 24, 25])
h, sub_f, sub_n = partition.extract_subgrid(g, c)
# There are 20 nodes in each layer
nodes_0 = np.array([0, 1, 2, 5, 6, 7, 10, 11])
nodes_1 = nodes_0 + 20
nodes_2 = np.array([0, 1, 2, 5, 6, 7]) + 40
nodes_3 = nodes_2 + 20
true_nodes = np.hstack((nodes_0, nodes_1, nodes_2, nodes_3))
faces_x = np.array([0, 1, 2, 5, 6, 15, 16, 30, 31, 32])
# y-faces start on 45
faces_y = np.array([45, 46, 49, 50, 53, 61, 65, 77, 78, 81, 82])
# z-faces start on 93
faces_z = np.array([93, 94, 97, 105, 106, 109, 117, 118, 129, 130])
true_faces = np.hstack((faces_x, faces_y, faces_z))
assert np.array_equal(true_nodes, sub_n)
assert np.array_equal(true_faces, sub_f)
self.compare_grid_geometries(g, h, c, true_faces, true_nodes)
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.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 = 0.5 * 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_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_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_subcell_topology_2d_cart_1(self):
x = np.ones(2, dtype=int)
g = structured.CartGrid(x)
subcell_topology = fvutils.SubcellTopology(g)
self.assertTrue(np.all(subcell_topology.cno == 0))
ncum = np.bincount(
subcell_topology.nno, weights=np.ones(subcell_topology.nno.size)
)
self.assertTrue(np.all(ncum == 2))
fcum = np.bincount(
subcell_topology.fno, weights=np.ones(subcell_topology.fno.size)
)
self.assertTrue(np.all(fcum == 2))
# There is only one cell, thus only unique subfno
usubfno = np.unique(subcell_topology.subfno)
self.assertTrue(usubfno.size == subcell_topology.subfno.size)
self.assertTrue(
np.all(np.in1d(subcell_topology.subfno, subcell_topology.subhfno))
)
def setUp(self):
self.tol = set_tol()
nc = np.array([2, 2])
g = structured.CartGrid(nc)
g.nodes[0, 4] = 1.5
g.compute_geometry()
self.g = g
def darcy_dualVEM_example0(**kwargs):
#######################
# Simple 2d Darcy problem with known exact solution
#######################
Nx = Ny = 25
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = tensor.SecondOrderTensor(g.dim, kxx)
f = np.ones(g.num_cells)
b_faces = g.get_all_boundary_faces()
bnd = bc.BoundaryCondition(g, b_faces, ["dir"] * b_faces.size)
bnd_val = np.zeros(g.num_faces)
solver = dual.DualVEM()
data = {"k": perm, "source": f, "bc": bnd, "bc_val": bnd_val}
D, rhs = solver.matrix_rhs(g, data)
up = sps.linalg.spsolve(D, rhs)
u, p = solver.extract_u(g, up), solver.extract_p(g, up)
P0u = solver.project_u(g, u)
if kwargs["visualize"]:
plot_grid(g, p, P0u)
norms = np.array([error.norm_L2(g, p), error.norm_L2(g, P0u)])
norms_known = np.array([0.041554943620853595, 0.18738227880674516])
assert np.allclose(norms, norms_known)
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_2d_cart_surf_discharge_positive(self):
g = structured.CartGrid([3, 2], [1, 1])
R = cg.rot(np.pi / 4., [0, 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 = 0.5 * 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 / 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 darcy_dual_hybridVEM_example0(**kwargs):
#######################
# Simple 2d Darcy problem with known exact solution
#######################
Nx = Ny = 25
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = tensor.SecondOrderTensor(g.dim, kxx)
f = np.ones(g.num_cells)
b_faces = g.get_all_boundary_faces()
bnd = bc.BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
bnd_val = np.zeros(g.num_faces)
solver = hybrid.HybridDualVEM()
data = {'perm': perm, 'source': f, 'bc': bnd, 'bc_val': bnd_val}
H, rhs = solver.matrix_rhs(g, data)
l = sps.linalg.spsolve(H, rhs)
u, p = solver.compute_up(g, l, data)
P0u = dual.DualVEM().project_u(g, u)
if kwargs['visualize']:
plot_grid(g, p, P0u)
norms = np.array([error.norm_L2(g, p), error.norm_L2(g, P0u)])
norms_known = np.array([0.041554943620853595, 0.18738227880674516])
assert np.allclose(norms, norms_known)
def test_create_partition_2d_cart(self):
g = structured.CartGrid([5, 5])
g.compute_geometry()
part = co.create_partition(co.tpfa_matrix(g))
known = np.array([0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 3, 2, 2, 2, 1, 3, 3, 2,\
4, 4, 3, 3, 4, 4, 4])
assert np.array_equal(part, known)
def make_grid(grid, grid_dims, domain, dim):
if grid.lower() == "cart" or grid.lower() == "cartesian":
return structured.CartGrid(grid_dims, domain)
elif (grid.lower() == "simplex" and dim == 2) or grid.lower() == "triangular":
return simplex.StructuredTriangleGrid(grid_dims, domain)
elif grid.lower() == "tetrahedral":
g = simplex.StructuredTetrahedralGrid(grid_dims, domain)
return g
def setup_cart_2d(nx):
g = structured.CartGrid(nx)
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = tensor.SecondOrderTensor(g.dim, kxx)
return g, perm
def darcy_dual_hybridVEM_example3(**kwargs):
#######################
# Simple 3d Darcy problem with known exact solution
#######################
Nx = Ny = Nz = 7
g = structured.CartGrid([Nx, Ny, Nz], [1, 1, 1])
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = tensor.SecondOrderTensor(g.dim, kxx)
def funP_ex(pt):
return (np.sin(2 * np.pi * pt[0]) * np.sin(2 * np.pi * pt[1]) *
np.sin(2 * np.pi * pt[2]))
def funU_ex(pt):
return [
-2 * np.pi * np.cos(2 * np.pi * pt[0]) *
np.sin(2 * np.pi * pt[1]) * np.sin(2 * np.pi * pt[2]),
-2 * np.pi * np.sin(2 * np.pi * pt[0]) *
np.cos(2 * np.pi * pt[1]) * np.sin(2 * np.pi * pt[2]),
-2 * np.pi * np.sin(2 * np.pi * pt[0]) *
np.sin(2 * np.pi * pt[1]) * np.cos(2 * np.pi * pt[2]),
]
def fun(pt):
return 12 * np.pi**2 * funP_ex(pt)
f = np.array([fun(pt) for pt in g.cell_centers.T])
b_faces = g.get_all_boundary_faces()
bnd = bc.BoundaryCondition(g, b_faces, ["dir"] * b_faces.size)
bnd_val = np.zeros(g.num_faces)
bnd_val[b_faces] = funP_ex(g.face_centers[:, b_faces])
solver = hybrid.HybridDualVEM()
data = {"perm": perm, "source": f, "bc": bnd, "bc_val": bnd_val}
H, rhs = solver.matrix_rhs(g, data)
l = sps.linalg.spsolve(H, rhs)
u, p = solver.compute_up(g, l, data)
P0u = dual.DualVEM().project_u(g, u)
if kwargs["visualize"]:
plot_grid(g, p, P0u)
p_ex = error.interpolate(g, funP_ex)
u_ex = error.interpolate(g, funU_ex)
np.set_printoptions(linewidth=999999)
np.set_printoptions(precision=16)
errors = np.array(
[error.error_L2(g, p, p_ex),
error.error_L2(g, P0u, u_ex)])
errors_known = np.array([0.1010936831876412, 0.0680593765009036])
assert np.allclose(errors, errors_known)
def test_tpfa_cart_2d():
""" Apply TPFA on Cartesian grid, should obtain Laplacian stencil. """
# Set up 3 X 3 Cartesian grid
nx = np.array([3, 3])
g = structured.CartGrid(nx)
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = tensor.SecondOrder(g.dim, kxx)
bound_faces = np.array([0, 3, 12])
bound = bc.BoundaryCondition(g, bound_faces, ['dir'] * bound_faces.size)
discr = tpfa.Tpfa()
d = _assign_params(g, perm, bound)
discr.discretize(g, d)
trm, bound_flux = d['flux'], d['bound_flux']
div = g.cell_faces.T
a = div * trm
b = -(div * bound_flux).A
# Checks on interior cell
mid = 4
assert a[mid, mid] == 4
assert a[mid - 1, mid] == -1
assert a[mid + 1, mid] == -1
assert a[mid - 3, mid] == -1
assert a[mid + 3, mid] == -1
assert np.all(b[mid, :] == 0)
# The first cell should have two Dirichlet bnds
assert a[0, 0] == 6
assert a[0, 1] == -1
assert a[0, 3] == -1
assert b[0, 0] == 2
assert b[0, 12] == 2
# Cell 3 has one Dirichlet, one Neumann face
print(a)
assert a[2, 2] == 4
assert a[2, 1] == -1
assert a[2, 5] == -1
assert b[2, 3] == 2
assert b[2, 14] == -1
# Cell 2 has one Neumann face
assert a[1, 1] == 3
assert a[1, 0] == -1
assert a[1, 2] == -1
assert a[1, 4] == -1
assert b[1, 13] == -1
return a
def make_grid(grid, grid_dims, domain, dim):
if grid.lower() == 'cart' or grid.lower() == 'cartesian':
return structured.CartGrid(grid_dims, domain)
elif (grid.lower() == 'simplex' and dim == 2) \
or grid.lower() == 'triangular':
return simplex.StructuredTriangleGrid(grid_dims, domain)
elif grid.lower() == 'tetrahedral':
g =simplex.StructuredTetrahedralGrid(grid_dims, domain)
return g
def test_tag_1d(self):
g = structured.CartGrid(3, 1)
self.assertTrue(np.array_equal(g.tags["fracture_faces"], [False] * g.num_faces))
self.assertTrue(np.array_equal(g.tags["fracture_nodes"], [False] * g.num_nodes))
self.assertTrue(np.array_equal(g.tags["tip_faces"], [False] * g.num_faces))
self.assertTrue(np.array_equal(g.tags["tip_nodes"], [False] * g.num_nodes))
known = [True, False, False, True]
self.assertTrue(np.array_equal(g.tags["domain_boundary_faces"], known))
self.assertTrue(np.array_equal(g.tags["domain_boundary_nodes"], known))
def test_create_partition_3d_cart(self):
g = structured.CartGrid([4, 4, 4])
g.compute_geometry()
part = co.create_partition(co.tpfa_matrix(g))
known = np.array([1, 1, 1, 1, 2, 4, 1, 3, 2, 2, 3, 3, 2, 2, 3, 3, 5, 4,\
1, 6, 4, 4, 4, 3, 2, 4, 7,
3, 8, 8, 3, 3, 5, 5, 6, 6, 5, 4, 7, 6, 8, 7, 7, 7, 8,\
8, 7, 9, 5, 5, 6, 6, 5, 5,
6, 6, 8, 8, 7, 9, 8, 8, 9, 9]) - 1
assert np.array_equal(part, known)
def coarsening_example2(**kwargs):
#######################
# Simple 2d coarsening based on tpfa for Cartesian grids
# anisotropic permeability
#######################
Nx = Ny = 7
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
if kwargs["visualize"]:
plot_grid(g, info="all", alpha=0)
kxx = 3 * np.ones(g.num_cells)
kyy = np.ones(g.num_cells)
perm = tensor.SecondOrderTensor(g.dim, kxx=kxx, kyy=kyy)
part = create_partition(tpfa_matrix(g, perm=perm))
g = generate_coarse_grid(g, part)
g.compute_geometry(is_starshaped=True)
if kwargs["visualize"]:
plot_grid(g, info="all", alpha=0)
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
part = create_partition(tpfa_matrix(g, perm=perm), cdepth=3)
g = generate_coarse_grid(g, part)
g.compute_geometry(is_starshaped=True)
if kwargs["visualize"]:
plot_grid(g, info="all", alpha=0)
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
part = create_partition(tpfa_matrix(g, perm=perm), cdepth=2, epsilon=1e-2)
g = generate_coarse_grid(g, part)
g.compute_geometry(is_starshaped=True)
if kwargs["visualize"]:
plot_grid(g, info="all", alpha=0)
def test_3d_coarse_dims_specified(self):
g = structured.CartGrid([4, 4, 4])
coarse_dims = np.array([2, 2, 2])
p = partition.partition_structured(g, coarse_dims)
p_known = np.array([[0, 0, 1, 1], [0, 0, 1, 1], [2, 2, 3, 3],
[2, 2, 3, 3], [0, 0, 1, 1], [0, 0, 1, 1],
[2, 2, 3, 3], [2, 2, 3, 3], [4, 4, 5, 5],
[4, 4, 5, 5], [6, 6, 7, 7], [6, 6, 7, 7],
[4, 4, 5, 5], [4, 4, 5, 5], [6, 6, 7, 7],
[6, 6, 7, 7]]).ravel('C')
assert np.allclose(p, p_known)
def test_2d_coarse_dims_specified(self):
g = structured.CartGrid([4, 10])
coarse_dims = np.array([2, 3])
p = partition.partition_structured(g, coarse_dims)
p_known = np.array(
[[0., 0., 1., 1.], [0., 0., 1., 1.], [0., 0., 1., 1.],
[2., 2., 3., 3.], [2., 2., 3., 3.], [2., 2., 3., 3.],
[4., 4., 5., 5.], [4., 4., 5., 5.], [4., 4., 5., 5.],
[4., 4., 5., 5.]],
dtype='int').ravel('C')
assert np.allclose(p, p_known)
def darcy_dualVEM_example3(**kwargs):
#######################
# Simple 3d Darcy problem with known exact solution
#######################
Nx = Ny = Nz = 7
g = structured.CartGrid([Nx, Ny, Nz], [1, 1, 1])
g.compute_geometry()
kxx = np.ones(g.num_cells)
perm = tensor.SecondOrder(g.dim, kxx)
def funP_ex(pt):
return np.sin(2*np.pi*pt[0])*np.sin(2*np.pi*pt[1])\
* np.sin(2*np.pi*pt[2])
def funU_ex(pt):
return [-2*np.pi*np.cos(2*np.pi*pt[0])\
* np.sin(2*np.pi*pt[1])*np.sin(2*np.pi*pt[2]),
-2*np.pi*np.sin(2*np.pi*pt[0])\
* np.cos(2*np.pi*pt[1])*np.sin(2*np.pi*pt[2]),
-2*np.pi*np.sin(2*np.pi*pt[0])\
* np.sin(2*np.pi*pt[1])*np.cos(2*np.pi*pt[2])]
def fun(pt):
return 12 * np.pi**2 * funP_ex(pt)
f = np.array([fun(pt) for pt in g.cell_centers.T])
b_faces = g.get_boundary_faces()
bnd = bc.BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
bnd_val = np.zeros(g.num_faces)
bnd_val[b_faces] = funP_ex(g.face_centers[:, b_faces])
solver = dual.DualVEM()
data = {'perm': perm, 'source': f, 'bc': bnd, 'bc_val': bnd_val}
D, rhs = solver.matrix_rhs(g, data)
up = sps.linalg.spsolve(D, rhs)
u, p = solver.extract_u(g, up), solver.extract_p(g, up)
P0u = solver.project_u(g, u)
if kwargs['visualize']:
plot_grid(g, p, P0u)
p_ex = error.interpolate(g, funP_ex)
u_ex = error.interpolate(g, funU_ex)
errors = np.array(
[error.error_L2(g, p, p_ex),
error.error_L2(g, P0u, u_ex)])
errors_known = np.array([0.1010936831876412, 0.0680593765009036])
assert np.allclose(errors, errors_known)
def test_create_partition_2d_cart_cdepth4(self):
g = structured.CartGrid([10, 10])
g.compute_geometry()
part = co.create_partition(co.tpfa_matrix(g), cdepth=4)
known = np.array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,\
1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1,\
1, 1, 1, 2, 2, 3, 3, 1, 1,
1, 1, 1, 2, 2, 2, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 3, 3,\
3, 3, 1, 1, 2, 2, 2, 2, 3,
3, 3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2,\
2]) - 1
assert np.array_equal(part, known)