def test01_solve_2d(self):
"""
Solve a simple 2D problem with no hanging nodes
"""
mesh = QuadMesh(resolution=(5, 5))
# Mark dirichlet boundaries
mesh.mark_region('left',
lambda x, dummy: np.abs(x) < 1e-9,
entity_type='half_edge')
mesh.mark_region('right',
lambda x, dummy: np.abs(x - 1) < 1e-9,
entity_type='half_edge')
Q1 = QuadFE(mesh.dim(), 'Q1')
dQ1 = DofHandler(mesh, Q1)
dQ1.distribute_dofs()
phi = Basis(dQ1, 'u')
phi_x = Basis(dQ1, 'ux')
phi_y = Basis(dQ1, 'uy')
problem = [
Form(1, test=phi_x, trial=phi_x),
Form(1, test=phi_y, trial=phi_y),
Form(0, test=phi)
]
assembler = Assembler(problem, mesh)
assembler.add_dirichlet('left', dir_fn=0)
assembler.add_dirichlet('right', dir_fn=1)
assembler.assemble()
# Get matrix dirichlet correction and right hand side
A = assembler.get_matrix().toarray()
x0 = assembler.assembled_bnd()
b = assembler.get_vector()
ua = np.zeros((phi.n_dofs(), 1))
int_dofs = assembler.get_dofs('interior')
ua[int_dofs, 0] = np.linalg.solve(A, b - x0)
dir_bc = assembler.get_dirichlet()
dir_vals = np.array([dir_bc[dof] for dof in dir_bc])
dir_dofs = [dof for dof in dir_bc]
ua[dir_dofs] = dir_vals
ue_fn = Nodal(f=lambda x: x[:, 0], basis=phi)
ue = ue_fn.data()
self.assertTrue(np.allclose(ue, ua))
self.assertTrue(np.allclose(x0 + A.dot(ua[int_dofs, 0]), b))
def test_edge_integrals(self):
"""
Test computing
"""
mesh = QuadMesh(resolution=(1, 1))
Q = QuadFE(2, 'Q1')
dQ = DofHandler(mesh, Q)
dQ.distribute_dofs()
phi = Basis(dQ, 'u')
f = Nodal(data=np.ones((phi.n_dofs(), 1)), basis=phi)
kernel = Kernel(f)
form = Form(kernel, dmu='ds')
assembler = Assembler(form, mesh)
cell = mesh.cells.get_leaves()[0]
shape_info = assembler.shape_info(cell)
xg, wg, phi, dofs = assembler.shape_eval(cell)
def test01_solve_1d(self):
"""
Test solving 1D systems
"""
mesh = Mesh1D(resolution=(20, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-9)
mesh.mark_region('right', lambda x: np.abs(x - 1) < 1e-9)
Q1 = QuadFE(1, 'Q1')
dQ1 = DofHandler(mesh, Q1)
dQ1.distribute_dofs()
phi = Basis(dQ1, 'u')
phi_x = Basis(dQ1, 'ux')
problem = [Form(1, test=phi_x, trial=phi_x), Form(0, test=phi)]
assembler = Assembler(problem, mesh)
assembler.add_dirichlet('left', dir_fn=0)
assembler.add_dirichlet('right', dir_fn=1)
assembler.assemble()
# Get matrix dirichlet correction and right hand side
A = assembler.get_matrix().toarray()
x0 = assembler.assembled_bnd()
b = assembler.get_vector()
ua = np.zeros((phi.n_dofs(), 1))
int_dofs = assembler.get_dofs('interior')
ua[int_dofs, 0] = np.linalg.solve(A, b - x0)
dir_bc = assembler.get_dirichlet()
dir_vals = np.array([dir_bc[dof] for dof in dir_bc])
dir_dofs = [dof for dof in dir_bc]
ua[dir_dofs] = dir_vals
ue_fn = Nodal(f=lambda x: x[:, 0], basis=phi)
ue = ue_fn.data()
self.assertTrue(np.allclose(ue, ua))
self.assertTrue(np.allclose(x0 + A.dot(ua[int_dofs, 0]), b))
vx = Explicit(f=lambda x: -x[:, 1], dim=2)
vy = Explicit(f=lambda x: x[:, 0], dim=2)
uB = Explicit(f=lambda x: np.cos(2 * np.pi * (x[:, 1] + 0.25)), dim=2)
problem = [
Form(epsilon, trial=ux, test=ux),
Form(epsilon, trial=uy, test=uy),
Form(vx, trial=ux, test=u),
Form(vy, trial=uy, test=u),
Form(0, test=u)
]
print('assembling', end=' ')
assembler = Assembler(problem, mesh)
assembler.assemble()
print('done')
print('solving', end=' ')
A = assembler.get_matrix()
b = np.zeros(u.n_dofs())
system = LS(u, A=A, b=b)
system.add_dirichlet_constraint('B', uB)
system.add_dirichlet_constraint('perimeter', 0)
system.solve_system()
ua = system.get_solution()
print('done')
plot = Plot()
plot.wire(ua)
def test_ft():
plot = Plot()
vb = Verbose()
# =============================================================================
# Parameters
# =============================================================================
#
# Flow
#
# permeability field
phi = Constant(1) # porosity
D = Constant(0.0252) # dispersivity
K = Constant(1) # permeability
# =============================================================================
# Mesh and Elements
# =============================================================================
# Mesh
mesh = QuadMesh(resolution=(30, 30))
# Mark left and right regions
mesh.mark_region('left',
lambda x, y: np.abs(x) < 1e-9,
entity_type='half_edge')
mesh.mark_region('right',
lambda x, y: np.abs(x - 1) < 1e-9,
entity_type='half_edge')
# Elements
p_element = QuadFE(2, 'Q1') # element for pressure
c_element = QuadFE(2, 'Q1') # element for concentration
# Dofhandlers
p_dofhandler = DofHandler(mesh, p_element)
c_dofhandler = DofHandler(mesh, c_element)
p_dofhandler.distribute_dofs()
c_dofhandler.distribute_dofs()
# Basis functions
p_ux = Basis(p_dofhandler, 'ux')
p_uy = Basis(p_dofhandler, 'uy')
p_u = Basis(p_dofhandler, 'u')
p_inflow = lambda x, y: np.ones(shape=x.shape)
p_outflow = lambda x, y: np.zeros(shape=x.shape)
c_inflow = lambda x, y: np.zeros(shape=x.shape)
# =============================================================================
# Solve the steady state flow equations
# =============================================================================
vb.comment('Solving flow equations')
# Define problem
flow_problem = [
Form(1, test=p_ux, trial=p_ux),
Form(1, test=p_uy, trial=p_uy),
Form(0, test=p_u)
]
# Assemble
vb.tic('assembly')
assembler = Assembler(flow_problem)
assembler.add_dirichlet('left', 1)
assembler.add_dirichlet('right', 0)
assembler.assemble()
vb.toc()
# Solve linear system
vb.tic('solve')
A = assembler.get_matrix().tocsr()
b = assembler.get_vector()
x0 = assembler.assembled_bnd()
# Interior nodes
pa = np.zeros((p_u.n_dofs(), 1))
int_dofs = assembler.get_dofs('interior')
pa[int_dofs, 0] = spla.spsolve(A, b - x0)
# Resolve Dirichlet conditions
dir_dofs, dir_vals = assembler.get_dirichlet(asdict=False)
pa[dir_dofs] = dir_vals
vb.toc()
# Pressure function
pfn = Nodal(data=pa, basis=p_u)
px = pfn.differentiate((1, 0))
py = pfn.differentiate((1, 1))
#plot.contour(px)
#plt.show()
# =============================================================================
# Transport Equations
# =============================================================================
# Specify initial condition
c0 = Constant(1)
dt = 1e-1
T = 6
N = int(np.ceil(T / dt))
c = Basis(c_dofhandler, 'c')
cx = Basis(c_dofhandler, 'cx')
cy = Basis(c_dofhandler, 'cy')
print('assembling transport equations')
k_phi = Kernel(f=phi)
k_advx = Kernel(f=[K, px], F=lambda K, px: -K * px)
k_advy = Kernel(f=[K, py], F=lambda K, py: -K * py)
tht = 1
m = [Form(kernel=k_phi, test=c, trial=c)]
s = [
Form(kernel=k_advx, test=c, trial=cx),
Form(kernel=k_advy, test=c, trial=cy),
Form(kernel=Kernel(D), test=cx, trial=cx),
Form(kernel=Kernel(D), test=cy, trial=cy)
]
problems = [m, s]
assembler = Assembler(problems)
assembler.add_dirichlet('left', 0, i_problem=0)
assembler.add_dirichlet('left', 0, i_problem=1)
assembler.assemble()
x0 = assembler.assembled_bnd()
# Interior nodes
int_dofs = assembler.get_dofs('interior')
# Dirichlet conditions
dir_dofs, dir_vals = assembler.get_dirichlet(asdict=False)
# System matrices
M = assembler.get_matrix(i_problem=0)
S = assembler.get_matrix(i_problem=1)
# Initialize c0 and cp
c0 = np.ones((c.n_dofs(), 1))
cp = np.zeros((c.n_dofs(), 1))
c_fn = Nodal(data=c0, basis=c)
#
# Compute solution
#
print('time stepping')
for i in range(N):
# Build system
A = M + tht * dt * S
b = M.dot(c0[int_dofs]) - (1 - tht) * dt * S.dot(c0[int_dofs])
# Solve linear system
cp[int_dofs, 0] = spla.spsolve(A, b)
# Add Dirichlet conditions
cp[dir_dofs] = dir_vals
# Record current iterate
c_fn.add_samples(data=cp)
# Update c0
c0 = cp.copy()
#plot.contour(c_fn, n_sample=i)
#
# Quantity of interest
#
def F(c, px, py, entity=None):
"""
Compute c(x,y,t)*(grad p * n)
"""
n = entity.unit_normal()
return c * (px * n[0] + py * n[1])
px.set_subsample(i=np.arange(41))
py.set_subsample(i=np.arange(41))
#kernel = Kernel(f=[c_fn,px,py], F=F)
kernel = Kernel(c_fn)
#print(kernel.n_subsample())
form = Form(kernel, flag='right', dmu='ds')
assembler = Assembler(form, mesh=mesh)
assembler.assemble()
QQ = assembler.assembled_forms()[0].aggregate_data()['array']
Q = np.array([assembler.get_scalar(i_sample=i) for i in np.arange(N + 1)])
t = np.linspace(0, T, N + 1)
plt.plot(t, Q)
plt.show()
print(Q)
def sample_qoi(q, dofhandler, return_state=False):
"""
Compute the Quantity of Interest
J(u) = -exp(q(1))*u'(1),
where u solves
-d/dx ( exp(q)* du/dx) = 1
u(0) = 0, u(1) = 1
for a sample of q's.
Inputs:
q: Nodal, (n_dofs, n_samples) function representing the log porosity
dofhandler: DofHandler
"""
# Basis
phi = Basis(dofhandler, 'v')
phi_x = Basis(dofhandler, 'vx')
n_dofs = phi.n_dofs()
# Define problem
expq_fn = Nodal(data=np.exp(q), basis=phi)
problem = [[Form(expq_fn, test=phi_x, trial=phi_x),
Form(1, test=phi)],
[Form(expq_fn, test=phi_x, dmu='dv', flag='right')]]
# Define assembler
assembler = Assembler(problem, dofhandler.mesh)
# Incorporate Dirichlet conditions
assembler.add_dirichlet('left', 0)
assembler.add_dirichlet('right', 1)
n_samples = expq_fn.n_subsample()
# Assemble system
assembler.assemble()
if return_state:
U = np.empty((n_dofs, n_samples))
J = np.zeros(n_samples)
for i in range(n_samples):
# Solve system
u = assembler.solve(i_problem=0, i_matrix=i, i_vector=0)
# Compute quantity of interest
J[i] = -u.dot(assembler.get_vector(1, i))
if return_state:
U[:, i] = u
if return_state:
return J, U
else:
return J
def test01_finite_elements():
"""
Test accuracy of the finite element approximation
"""
#
# Construct reference solution
#
plot = Plot(quickview=False)
# Mesh
mesh = Mesh1D(resolution=(2**11, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-10)
mesh.mark_region('right', lambda x: np.abs(x - 1) < 1e-10)
# Element
Q1 = QuadFE(mesh.dim(), 'Q1')
dQ1 = DofHandler(mesh, Q1)
dQ1.distribute_dofs()
# Basis
phi = Basis(dQ1, 'v')
phi_x = Basis(dQ1, 'vx')
#
# Covariance
#
cov = Covariance(dQ1, name='gaussian', parameters={'l': 0.05})
cov.compute_eig_decomp()
lmd, V = cov.get_eig_decomp()
d = len(lmd)
#
# Sample and plot full dimensional parameter and solution
#
n_samples = 1
z = np.random.randn(d, n_samples)
q_ref = sample_q0(V, lmd, d, z)
print(q_ref.shape)
# Define finite element function
q_ref_fn = Nodal(data=q_ref, basis=phi)
problem = [[Form(q_ref_fn, test=phi_x, trial=phi_x),
Form(1, test=phi)],
[Form(q_ref_fn, test=phi_x, dmu='dv', flag='right')]]
# Define assembler
assembler = Assembler(problem)
# Incorporate Dirichlet conditions
assembler.add_dirichlet('left', 0)
assembler.add_dirichlet('right', 1)
# Assemble system
assembler.assemble()
# Solve system
u_ref = assembler.solve()
# Compute quantity of interest
J_ref = u_ref.dot(assembler.get_vector(1))
# Plot
fig = plt.figure(figsize=(6, 4))
ax = fig.add_subplot(111)
u_ref_fn = Nodal(basis=phi, data=u_ref)
ax = plot.line(u_ref_fn, axis=ax)
n_levels = 10
J = np.zeros(n_levels)
for l in range(10):
comment.comment('level: %d' % (l))
#
# Mesh
#
mesh = Mesh1D(resolution=(2**l, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-10)
mesh.mark_region('right', lambda x: np.abs(x - 1) < 1e-10)
#
# Element
#
Q1 = QuadFE(mesh.dim(), 'Q1')
dQ1 = DofHandler(mesh, Q1)
dQ1.distribute_dofs()
#
# Basis
#
phi = Basis(dQ1, 'v')
phi_x = Basis(dQ1, 'vx')
# Define problem
problem = [[
Form(q_ref_fn, test=phi_x, trial=phi_x),
Form(1, test=phi)
], [Form(q_ref_fn, test=phi_x, dmu='dv', flag='right')]]
assembler = Assembler(problem)
# Incorporate Dirichlet conditions
assembler.add_dirichlet('left', 0)
assembler.add_dirichlet('right', 1)
assembler.assemble()
A = assembler.get_matrix()
print('A shape', A.shape)
u = assembler.solve()
J[l] = u.dot(assembler.get_vector(1))
print(u.shape)
print(phi.n_dofs())
ufn = Nodal(basis=phi, data=u)
ax = plot.line(ufn, axis=ax)
plt.show()
#
# Plots
#
# Formatting
plt.rc('text', usetex=True)
# Figure sizes
fs2 = (3, 2)
fs1 = (4, 3)
print(J_ref)
print(J)
#
# Plot truncation error for mean and variance of J
#
fig = plt.figure(figsize=fs2)
ax = fig.add_subplot(111)
err = np.array([np.abs(J[i] - J_ref) for i in range(n_levels)])
h = np.array([2**(-l) for l in range(n_levels)])
plt.loglog(h, err, '.-')
ax.set_xlabel(r'$h$')
ax.set_ylabel(r'$|J-J^h|$')
plt.tight_layout()
fig.savefig('fig/ex02_gauss_fem_error.eps')
def test03_solve_2d(self):
"""
Test problem with Neumann conditions
"""
#
# Define Mesh
#
mesh = QuadMesh(resolution=(2, 1))
mesh.cells.get_child(1).mark(1)
mesh.cells.refine(refinement_flag=1)
# Mark left and right boundaries
bm_left = lambda x, dummy: np.abs(x) < 1e-9
bm_right = lambda x, dummy: np.abs(1 - x) < 1e-9
mesh.mark_region('left', bm_left, entity_type='half_edge')
mesh.mark_region('right', bm_right, entity_type='half_edge')
for etype in ['Q1', 'Q2', 'Q3']:
#
# Define element and basis type
#
element = QuadFE(2, etype)
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
uy = Basis(dofhandler, 'uy')
#
# Exact solution
#
ue = Nodal(f=lambda x: x[:, 0], basis=u)
#
# Set up forms
#
one = Constant(1)
ax = Form(kernel=Kernel(one), trial=ux, test=ux)
ay = Form(kernel=Kernel(one), trial=uy, test=uy)
L = Form(kernel=Kernel(Constant(0)), test=u)
Ln = Form(kernel=Kernel(one), test=u, dmu='ds', flag='right')
problem = [ax, ay, L, Ln]
assembler = Assembler(problem, mesh)
assembler.add_dirichlet('left', dir_fn=0)
assembler.add_hanging_nodes()
assembler.assemble()
#
# Automatic solve
#
ya = assembler.solve()
self.assertTrue(np.allclose(ue.data()[:, 0], ya))
#
# Explicit solve
#
# System Matrices
A = assembler.get_matrix().toarray()
b = assembler.get_vector()
x0 = assembler.assembled_bnd()
# Solve linear system
xa = np.zeros(u.n_dofs())
int_dofs = assembler.get_dofs('interior')
xa[int_dofs] = np.linalg.solve(A, b - x0)
# Resolve Dirichlet conditions
dir_dofs, dir_vals = assembler.get_dirichlet(asdict=False)
xa[dir_dofs] = dir_vals[:, 0]
# Resolve hanging nodes
C = assembler.hanging_node_matrix()
xa += C.dot(xa)
self.assertTrue(np.allclose(ue.data()[:, 0], xa))
def test02_solve_2d(self):
"""
Solve 2D problem with hanging nodes
"""
# Mesh
mesh = QuadMesh(resolution=(2, 2))
mesh.cells.get_leaves()[0].mark(0)
mesh.cells.refine(refinement_flag=0)
mesh.mark_region('left',
lambda x, y: abs(x) < 1e-9,
entity_type='half_edge')
mesh.mark_region('right',
lambda x, y: abs(x - 1) < 1e-9,
entity_type='half_edge')
# Element
Q1 = QuadFE(2, 'Q1')
dofhandler = DofHandler(mesh, Q1)
dofhandler.distribute_dofs()
dofhandler.set_hanging_nodes()
# Basis functions
phi = Basis(dofhandler, 'u')
phi_x = Basis(dofhandler, 'ux')
phi_y = Basis(dofhandler, 'uy')
#
# Define problem
#
problem = [
Form(1, trial=phi_x, test=phi_x),
Form(1, trial=phi_y, test=phi_y),
Form(0, test=phi)
]
ue = Nodal(f=lambda x: x[:, 0], basis=phi)
xe = ue.data().ravel()
#
# Assemble without Dirichlet and without Hanging Nodes
#
assembler = Assembler(problem, mesh)
assembler.add_dirichlet('left', dir_fn=0)
assembler.add_dirichlet('right', dir_fn=1)
assembler.add_hanging_nodes()
assembler.assemble()
# Get dofs for different regions
int_dofs = assembler.get_dofs('interior')
# Get matrix and vector
A = assembler.get_matrix().toarray()
b = assembler.get_vector()
x0 = assembler.assembled_bnd()
# Solve linear system
xa = np.zeros(phi.n_dofs())
xa[int_dofs] = np.linalg.solve(A, b - x0)
# Resolve Dirichlet conditions
dir_dofs, dir_vals = assembler.get_dirichlet(asdict=False)
xa[dir_dofs] = dir_vals[:, 0]
# Resolve hanging nodes
C = assembler.hanging_node_matrix()
xa += C.dot(xa)
self.assertTrue(np.allclose(xa, xe))