def demo64(self, permeability, obs_case=1):
mesh = UnitSquareMesh(63, 63)
ak_values = permeability
flux_order = 3
s = scenarios.darcy_problem_1()
DRT = fenics.FiniteElement("DRT", mesh.ufl_cell(), flux_order)
# Lagrange
CG = fenics.FiniteElement("CG", mesh.ufl_cell(), flux_order + 1)
if s.integral_constraint:
# From https://fenicsproject.org/qa/14184/how-to-solve-linear-system-with-constaint
R = fenics.FiniteElement("R", mesh.ufl_cell(), 0)
W = fenics.FunctionSpace(mesh, fenics.MixedElement([DRT, CG, R]))
# Define trial and test functions
(sigma, u, r) = fenics.TrialFunctions(W)
(tau, v, r_) = fenics.TestFunctions(W)
else:
W = fenics.FunctionSpace(mesh, DRT * CG)
# Define trial and test functions
(sigma, u) = fenics.TrialFunctions(W)
(tau, v) = fenics.TestFunctions(W)
f = s.source_function
g = s.g
# Define property field function
W_CG = fenics.FunctionSpace(mesh, "Lagrange", 1)
if s.ak is None:
ak = property_field.get_conductivity(W_CG, ak_values)
else:
ak = s.ak
# Define variational form
a = (fenics.dot(sigma, tau) + fenics.dot(ak * fenics.grad(u), tau) +
fenics.dot(sigma, fenics.grad(v))) * fenics.dx
L = -f * v * fenics.dx + g * v * fenics.ds
# L = 0
if s.integral_constraint:
# Lagrange multiplier? See above link.
a += r_ * u * fenics.dx + v * r * fenics.dx
# Define Dirichlet BC
bc = fenics.DirichletBC(W.sub(1), s.dirichlet_bc, s.gamma_dirichlet)
# Compute solution
w = fenics.Function(W)
fenics.solve(a == L, w, bc)
# fenics.solve(a == L, w)
if s.integral_constraint:
(sigma, u, r) = w.split()
else:
(sigma, u) = w.split()
x = u.compute_vertex_values(mesh)
x2 = sigma.compute_vertex_values(mesh)
p = x
pre = p.reshape((64, 64))
vx = x2[:4096].reshape((64, 64))
vy = x2[4096:].reshape((64, 64))
if obs_case == 1:
dd = np.zeros([8, 8])
pos = np.full((8 * 8, 2), 0)
col = [4, 12, 20, 28, 36, 44, 52, 60]
position = [4, 12, 20, 28, 36, 44, 52, 60]
for i in range(8):
for j in range(8):
row = position
pos[8 * i + j, :] = [col[i], row[j]]
dd[i, j] = pre[col[i], row[j]]
like = dd.reshape(8 * 8, )
return like, pre, vx, vy, ak_values, pos
def demo16(self, permeability, obs_case=1):
"""This demo program solves the mixed formulation of Poisson's
equation:
sigma + grad(u) = 0 in Omega
div(sigma) = f in Omega
du/dn = g on Gamma_N
u = u_D on Gamma_D
The corresponding weak (variational problem)
<sigma, tau> + <grad(u), tau> = 0
for all tau
- <sigma, grad(v)> = <f, v> + <g, v>
for all v
is solved using DRT (Discontinuous Raviart-Thomas) elements
of degree k for (sigma, tau) and CG (Lagrange) elements
of degree k + 1 for (u, v) for k >= 1.
"""
mesh = UnitSquareMesh(15, 15)
ak_values = permeability
flux_order = 3
s = scenarios.darcy_problem_1()
DRT = fenics.FiniteElement("DRT", mesh.ufl_cell(), flux_order)
# Lagrange
CG = fenics.FiniteElement("CG", mesh.ufl_cell(), flux_order + 1)
if s.integral_constraint:
# From https://fenicsproject.org/qa/14184/how-to-solve-linear-system-with-constaint
R = fenics.FiniteElement("R", mesh.ufl_cell(), 0)
W = fenics.FunctionSpace(mesh, fenics.MixedElement([DRT, CG, R]))
# Define trial and test functions
(sigma, u, r) = fenics.TrialFunctions(W)
(tau, v, r_) = fenics.TestFunctions(W)
else:
W = fenics.FunctionSpace(mesh, DRT * CG)
# Define trial and test functions
(sigma, u) = fenics.TrialFunctions(W)
(tau, v) = fenics.TestFunctions(W)
f = s.source_function
g = s.g
# Define property field function
W_CG = fenics.FunctionSpace(mesh, "Lagrange", 1)
if s.ak is None:
ak = property_field.get_conductivity(W_CG, ak_values)
else:
ak = s.ak
# Define variational form
a = (fenics.dot(sigma, tau) + fenics.dot(ak * fenics.grad(u), tau) +
fenics.dot(sigma, fenics.grad(v))) * fenics.dx
L = -f * v * fenics.dx + g * v * fenics.ds
# L = 0
if s.integral_constraint:
# Lagrange multiplier? See above link.
a += r_ * u * fenics.dx + v * r * fenics.dx
# Define Dirichlet BC
bc = fenics.DirichletBC(W.sub(1), s.dirichlet_bc, s.gamma_dirichlet)
# Compute solution
w = fenics.Function(W)
fenics.solve(a == L, w, bc)
# fenics.solve(a == L, w)
if s.integral_constraint:
(sigma, u, r) = w.split()
else:
(sigma, u) = w.split()
x = u.compute_vertex_values(mesh)
x2 = sigma.compute_vertex_values(mesh)
p = x
pre = p.reshape((16, 16))
vx = x2[:256].reshape((16, 16))
vy = x2[256:].reshape((16, 16))
if obs_case == 1:
dd = np.zeros([8, 8])
pos = np.full((8 * 8, 2), 0)
col = [1, 3, 5, 7, 9, 11, 13, 15]
position = [1, 3, 5, 7, 9, 11, 13, 15]
for i in range(8):
for j in range(8):
row = position
pos[8 * i + j, :] = [col[i], row[j]]
dd[i, j] = pre[col[i], row[j]]
like = dd.reshape(8 * 8, )
return like, pre, vx, vy, ak_values, pos