def main(N):
Nx = Ny = N
#g = structured.CartGrid([Nx, Ny], [1, 1])
g = simplex.StructuredTriangleGrid([Nx, Ny], [1, 1])
R = cg.rot(np.pi / 4., [1, 0, 0])
g.nodes = np.dot(R, g.nodes)
g.compute_geometry(is_embedded=True)
#co.coarsen(g, 'by_volume')
# Assign parameters
data = add_data(g)
# Choose and define the solvers
solver_flow = vem_dual.DualVEM('flow')
A_flow, b_flow = solver_flow.matrix_rhs(g, data)
solver_source = vem_source.Integral('flow')
A_source, b_source = solver_source.matrix_rhs(g, data)
up = sps.linalg.spsolve(A_flow + A_source, b_flow + b_source)
u = solver_flow.extract_u(g, up)
p = solver_flow.extract_p(g, up)
P0u = solver_flow.project_u(g, u, data)
diam = np.amax(g.cell_diameters())
return diam, error_p(g, p)
def test_upwind_example2(self, if_export=False):
#######################
# Simple 2d upwind problem with explicit Euler scheme in time coupled with
# a Darcy problem
#######################
T = 2
Nx, Ny = 10, 10
folder = 'example2'
def funp_ex(pt):
return -np.sin(pt[0]) * np.sin(pt[1]) - pt[0]
g = structured.CartGrid([Nx, Ny], [1, 1])
g.compute_geometry()
param = Parameters(g)
# Permeability
perm = tensor.SecondOrder(g.dim, kxx=np.ones(g.num_cells))
param.set_tensor("flow", perm)
# Source term
param.set_source("flow", np.zeros(g.num_cells))
# Boundaries
b_faces = g.get_boundary_faces()
bc = BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
bc_val = np.zeros(g.num_faces)
bc_val[b_faces] = funp_ex(g.face_centers[:, b_faces])
param.set_bc("flow", bc)
param.set_bc_val("flow", bc_val)
# Darcy solver
data = {'param': param}
solver = vem_dual.DualVEM("flow")
D_flow, b_flow = solver.matrix_rhs(g, data)
solver_source = vem_source.Integral('flow')
D_source, b_source = solver_source.matrix_rhs(g, data)
up = sps.linalg.spsolve(D_flow + D_source, b_flow + b_source)
p, u = solver.extract_p(g, up), solver.extract_u(g, up)
P0u = solver.project_u(g, u, data)
save = Exporter(g, "darcy", folder)
if if_export:
save.write_vtk({'pressure': p, "P0u": P0u})
# Discharge
dis = u
# Boundaries
bc = BoundaryCondition(g, b_faces, ['dir'] * b_faces.size)
bc_val = np.hstack(([1], np.zeros(g.num_faces - 1)))
param.set_bc("transport", bc)
param.set_bc_val("transport", bc_val)
data = {'param': param, 'discharge': dis}
# Advect solver
advect = upwind.Upwind("transport")
U, rhs = advect.matrix_rhs(g, data)
data['deltaT'] = advect.cfl(g, data)
M, _ = mass_matrix.MassMatrix().matrix_rhs(g, data)
conc = np.zeros(g.num_cells)
M_minus_U = M - U
invM, _ = mass_matrix.InvMassMatrix().matrix_rhs(g, data)
# Loop over the time
Nt = int(T / data['deltaT'])
time = np.empty(Nt)
save.change_name("conc_darcy")
for i in np.arange(Nt):
# Update the solution
conc = invM.dot((M_minus_U).dot(conc) + rhs)
time[i] = data['deltaT'] * i
if if_export:
save.write_vtk({"conc": conc}, time_step=i)
if if_export:
save.write_pvd(time)
known = \
np.array([9.63168200e-01, 8.64054875e-01, 7.25390695e-01,
5.72228235e-01, 4.25640080e-01, 2.99387331e-01,
1.99574336e-01, 1.26276876e-01, 7.59011550e-02,
4.33431230e-02, 3.30416807e-02, 1.13058617e-01,
2.05372538e-01, 2.78382057e-01, 3.14035373e-01,
3.09920132e-01, 2.75024694e-01, 2.23163145e-01,
1.67386939e-01, 1.16897527e-01, 1.06111312e-03,
1.11951850e-02, 3.87907727e-02, 8.38516119e-02,
1.36617802e-01, 1.82773271e-01, 2.10446545e-01,
2.14651936e-01, 1.97681518e-01, 1.66549151e-01,
3.20751341e-05, 9.85780113e-04, 6.07062715e-03,
1.99393042e-02, 4.53237556e-02, 8.00799828e-02,
1.17199623e-01, 1.47761481e-01, 1.64729339e-01,
1.65390555e-01, 9.18585872e-07, 8.08267622e-05,
8.47227168e-04, 4.08879583e-03, 1.26336029e-02,
2.88705048e-02, 5.27841497e-02, 8.10459333e-02,
1.07956484e-01, 1.27665318e-01, 2.51295298e-08,
6.29844122e-06, 1.09361990e-04, 7.56743783e-04,
3.11384414e-03, 9.04446601e-03, 2.03443897e-02,
3.75208816e-02, 5.89595194e-02, 8.11457277e-02,
6.63498510e-10, 4.73075468e-07, 1.33728945e-05,
1.30243418e-04, 7.01905707e-04, 2.55272292e-03,
6.96686157e-03, 1.52290448e-02, 2.78607282e-02,
4.40402650e-02, 1.71197497e-11, 3.47118057e-08,
1.57974045e-06, 2.13489614e-05, 1.48634295e-04,
6.68104990e-04, 2.18444135e-03, 5.58646819e-03,
1.17334966e-02, 2.09744728e-02, 4.37822313e-13,
2.52373622e-09, 1.83589660e-07, 3.40553325e-06,
3.02948532e-05, 1.66504215e-04, 6.45119867e-04,
1.90731440e-03, 4.53436628e-03, 8.99977737e-03,
1.12627412e-14, 1.84486857e-10, 2.13562387e-08,
5.39492977e-07, 6.08223906e-06, 4.05535296e-05,
1.84731221e-04, 6.25871542e-04, 1.66459389e-03,
3.59980231e-03])
assert np.allclose(conc, known)
def source_disc(self):
if self.is_GridBucket:
return vem_source.IntegralMixedDim(physics=self.physics)
else:
return vem_source.Integral(physics=self.physics)