def test03_dJdq():
"""
Compute dJdq for a simple problem, check that it works
"""
#
# Mesh
#
mesh = Mesh1D(resolution=(20, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-10)
mesh.mark_region('right', lambda x: np.abs(x - 1) < 1e-10)
#
# Element
#
Q = QuadFE(mesh.dim(), 'Q3')
dh = DofHandler(mesh, Q)
dh.distribute_dofs()
nx = dh.n_dofs()
x = dh.get_dof_vertices()
#
# Basis
#
phi = Basis(dh, 'v')
phi_x = Basis(dh, 'vx')
#
# Parameters
#
# Reference q
q_ref = Nodal(data=np.zeros(nx), basis=phi)
# Perturbation
dq = Nodal(data=np.ones(nx), basis=phi)
#
# Sample Reference QoI
#
J, u_ref = sample_qoi(q_ref.data(), dh, return_state=True)
u_ref = Nodal(data=u_ref, basis=phi)
#
# Compute dJdq
#
# Perturbation method
Jp_per = dJdq_per(q_ref, dq, dh)
# Sensitivity method
Jp_sen = dJdq_sen(q_ref, u_ref, dq)
# Adjoint method
Jp_adj = dJdq_adj(q_ref, u_ref, dq)
# Check that the answers are close to -1
assert np.allclose(Jp_per, -1)
assert np.allclose(Jp_sen, -1)
assert np.allclose(Jp_adj, -1)
def sensitivity_sample_qoi(exp_q, dofhandler):
"""
Sample QoI by means of Taylor expansion
J(q+dq) ~= J(q) + dJdq(q)dq
"""
# Basis
phi = Basis(dofhandler, 'v')
phi_x = Basis(dofhandler, 'vx')
# Define problem
exp_q_fn = Nodal(data=exp_q, basis=phi)
primal = [Form(exp_q_fn, test=phi_x, trial=phi_x), Form(1, test=phi)]
adjoint = [Form(exp_q_fn, test=phi_x, trial=phi_x), Form(0, test=phi)]
qoi = [Form(exp_q_fn, test=phi_x)]
problems = [primal, adjoint, qoi]
# Define assembler
assembler = Assembler(problems)
#
# Dirichlet conditions for primal problem
#
assembler.add_dirichlet('left', 0, i_problem=0)
assembler.add_dirichlet('right', 1, i_problem=0)
# Dirichlet conditions for adjoint problem
assembler.add_dirichlet('left', 0, i_problem=1)
assembler.add_dirichlet('right', -1, i_problem=1)
# Assemble system
assembler.assemble()
# Compute solution and qoi at q (primal)
u = assembler.solve(i_problem=0)
# Compute solution of the adjoint problem
v = assembler.solve(i_problem=1)
# Evaluate J
J = u.dot(assembler.get_vector(2))
#
# Assemble gradient
#
ux_fn = Nodal(data=u, basis=phi_x)
vx_fn = Nodal(data=v, basis=phi_x)
k_int = Kernel(f=[exp_q_fn, ux_fn, vx_fn],
F=lambda exp_q, ux, vx: exp_q * ux * vx)
problem = [Form(k_int, test=phi)]
assembler = Assembler(problem)
assembler.assemble()
dJ = -assembler.get_vector()
return dJ
def test_set(self):
mesh = QuadMesh(resolution=(1, 1))
element = QuadFE(2, 'Q1')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
px = Basis(dofhandler, 'ux')
p = Basis(dofhandler, 'ux')
self.assertNotEqual(px, p)
def sample_qoi(q, dofhandler):
"""
Sample total energy of output for a given sample of q's
"""
#
# Set up weak form
#
# Basis
phi = Basis(dofhandler, 'v')
phi_x = Basis(dofhandler, 'vx')
# Elliptic problem
problems = [[Form(q, test=phi_x, trial=phi_x), Form(1, test=phi)],
[Form(1, test=phi, trial=phi)]]
# Assemble
assembler = Assembler(problems, mesh)
assembler.assemble()
# System matrices
A = assembler.af[0]['bilinear'].get_matrix()
b = assembler.af[0]['linear'].get_matrix()
M = assembler.af[1]['bilinear'].get_matrix()
# Define linear system
system = LS(phi)
system.add_dirichlet_constraint('left',1)
system.add_dirichlet_constraint('right',0)
n_samples = q.n_samples()
y_smpl = []
QoI_smpl = []
for i in range(n_samples):
# Sample system
if n_samples > 1:
Ai = A[i]
else:
Ai = A
system.set_matrix(Ai)
system.set_rhs(b.copy())
# Solve system
system.solve_system()
# Record solution and qoi
y = system.get_solution(as_function=False)
y_smpl.append(y)
QoI_smpl.append(y.T.dot(M.dot(y)))
# Convert to numpy array
y_smpl = np.concatenate(y_smpl,axis=1)
QoI = np.concatenate(QoI_smpl, axis=1).ravel()
return y_smpl, QoI
def test02_1d_dirichlet_higher_order(self):
mesh = Mesh1D()
for etype in ['Q2', 'Q3']:
element = QuadFE(1, etype)
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Basis functions
ux = Basis(dofhandler, 'ux')
u = Basis(dofhandler, 'u')
# Exact solution
ue = Nodal(f=lambda x: x * (1 - x), basis=u)
# Define coefficient functions
one = Constant(1)
two = Constant(2)
# Define forms
a = Form(kernel=Kernel(one), trial=ux, test=ux)
L = Form(kernel=Kernel(two), test=u)
problem = [a, L]
# Assemble problem
assembler = Assembler(problem, mesh)
assembler.assemble()
A = assembler.get_matrix()
b = assembler.get_vector()
# Set up linear system
system = LinearSystem(u, A=A, b=b)
# Boundary functions
bnd_left = lambda x: np.abs(x) < 1e-9
bnd_right = lambda x: np.abs(1 - x) < 1e-9
# Mark mesh
mesh.mark_region('left', bnd_left, entity_type='vertex')
mesh.mark_region('right', bnd_right, entity_type='vertex')
# Add Dirichlet constraints to system
system.add_dirichlet_constraint('left', 0)
system.add_dirichlet_constraint('right', 0)
# Solve system
system.solve_system()
system.resolve_constraints()
# Compare solution with the exact solution
ua = system.get_solution(as_function=True)
self.assertTrue(np.allclose(ua.data(), ue.data()))
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_constructor(self):
#
# Define mesh, element, and dofhandler
#
mesh = QuadMesh(box=[0, 20, 0, 20],
resolution=(20, 20),
periodic={0, 1})
dim = mesh.dim()
element = QuadFE(dim, 'Q2')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
basis = Basis(dofhandler, 'u')
alph = 2
kppa = 1
# Symmetric tensor gma T + bta* vv^T
gma = 0.1
bta = 25
p = lambda x: 10/np.pi*(0.75*np.sin(np.pi*x[:,0]/10)+\
0.25*np.sin(np.pi*x[:,1]/10))
f = Nodal(f=p, basis=basis)
fx = f.differentiate((1, 0))
fy = f.differentiate((1, 1))
#plot.contour(f)
x = np.linspace(0, 20, 12)
X, Y = np.meshgrid(x, x)
xy = np.array([X.ravel(), Y.ravel()]).T
U = fx.eval(xy).reshape(X.shape)
V = fy.eval(xy).reshape(X.shape)
v1 = lambda x: -0.25 * np.cos(np.pi * x[:, 1] / 10)
v2 = lambda x: 0.75 * np.cos(np.pi * x[:, 0] / 10)
U = v1(xy).reshape(X.shape)
V = v2(xy).reshape(X.shape)
#plt.quiver(X,Y, U, V)
#plt.show()
h11 = Explicit(lambda x: gma + bta * v1(x) * v1(x), dim=2)
h12 = Explicit(lambda x: bta * v1(x) * v2(x), dim=2)
h22 = Explicit(lambda x: gma + bta * v2(x) * v2(x), dim=2)
tau = (h11, h12, h22)
#tau = (Constant(2), Constant(1), Constant(1))
#
# Define default elliptic field
#
u = EllipticField(dofhandler, kappa=1, tau=tau, gamma=2)
Q = u.precision()
v = Nodal(data=u.sample(mode='precision', decomposition='chol'),
basis=basis)
plot = Plot(20)
plot.contour(v)
def experiment01():
"""
Compute the quantity of interest, it's expectation and variance
"""
#
# FE Discretization
#
# Computational mesh
mesh = Mesh1D(resolution=(64, ))
# Element
element = QuadFE(mesh.dim(), 'DQ0')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Linear Functional
mesh.mark_region('integrate',
lambda x: x > 0.75,
entity_type='cell',
strict_containment=False)
phi = Basis(dofhandler)
assembler = Assembler(Form(1, test=phi, flag='integrate'))
assembler.assemble()
L = assembler.get_vector()
def test_assemble_iiform(self):
mesh = Mesh1D(resolution=(1, ))
Q1 = QuadFE(1, 'DQ1')
dofhandler = DofHandler(mesh, Q1)
dofhandler.distribute_dofs()
phi = Basis(dofhandler, 'u')
k = Explicit(lambda x, y: x * y, n_variables=2, dim=1)
kernel = Kernel(k)
form = IIForm(kernel, test=phi, trial=phi)
assembler = Assembler(form, mesh)
assembler.assemble()
Ku = Nodal(lambda x: 1 / 3 * x, basis=phi)
#af = assembler.af[0]['bilinear']
M = assembler.get_matrix().toarray()
u = Nodal(lambda x: x, basis=phi)
u_vec = u.data()
self.assertTrue(np.allclose(M.dot(u_vec), Ku.data()))
def test_assemble_ipform(self):
# =====================================================================
# Test 7: Assemble Kernel
# =====================================================================
mesh = Mesh1D(resolution=(10, ))
Q1 = QuadFE(1, 'DQ1')
dofhandler = DofHandler(mesh, Q1)
dofhandler.distribute_dofs()
phi = Basis(dofhandler, 'u')
k = Explicit(lambda x, y: x * y, n_variables=2, dim=1)
kernel = Kernel(k)
form = IPForm(kernel, test=phi, trial=phi)
assembler = Assembler(form, mesh)
assembler.assemble()
#af = assembler.af[0]['bilinear']
M = assembler.get_matrix().toarray()
u = Nodal(lambda x: x, basis=phi)
v = Nodal(lambda x: 1 - x, basis=phi)
u_vec = u.data()
v_vec = v.data()
I = v_vec.T.dot(M.dot(u_vec))
self.assertAlmostEqual(I[0, 0], 1 / 18)
def sampling_error():
"""
Test the sampling error by comparing the accuracy of the quantities of
interest q1 = E[|y|] and q2 = E[y(0.5)]
"""
c = Verbose()
mesh = Mesh1D(resolution=(1026,))
mesh.mark_region('left', lambda x:np.abs(x)<1e-10)
mesh.mark_region('right', lambda x:np.abs(x-1)<1e-10)
element = QuadFE(1,'Q1')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
dofhandler.set_dof_vertices()
phi = Basis(dofhandler,'u')
phi_x = Basis(dofhandler,'ux')
ns_ref = 10000
z = get_points(n_samples=ns_ref)
q = set_diffusion(dofhandler,z)
problems = [[Form(q, test=phi_x, trial=phi_x), Form(1, test=phi)],
[Form(1, test=phi, trial=phi)]]
c.tic('assembling')
assembler = Assembler(problems, mesh)
assembler.assemble()
c.toc()
A = assembler.af[0]['bilinear'].get_matrix()
b = assembler.af[0]['linear'].get_matrix()
M = assembler.af[0]['bilinear'].get_matrix()
system = LS(phi)
system.add_dirichlet_constraint('left')
system.add_dirichlet_constraint('right')
c.tic('solving')
for n in range(ns_ref):
system.set_matrix(A[n])
system.set_rhs(b.copy())
system.solve_system()
c.toc()
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 sample_state(mesh,dQ,z,mflag,reference=False):
"""
Compute the sample output corresponding to a given input
"""
n_samples = z.shape[0]
q = set_diffusion(dQ,z)
phi = Basis(dQ,'u', mflag)
phi_x = Basis(dQ, 'ux', mflag)
if reference:
problems = [[Form(q,test=phi_x,trial=phi_x), Form(1,test=phi)],
[Form(1,test=phi, trial=phi)],
[Form(1,test=phi_x, trial=phi_x)]]
else:
problems = [[Form(q,test=phi_x,trial=phi_x), Form(1,test=phi)]]
assembler = Assembler(problems, mesh, subforest_flag=mflag)
assembler.assemble()
A = assembler.af[0]['bilinear'].get_matrix()
b = assembler.af[0]['linear'].get_matrix()
if reference:
M = assembler.af[1]['bilinear'].get_matrix()
K = assembler.af[2]['bilinear'].get_matrix()
system = LS(phi)
system.add_dirichlet_constraint('left')
system.add_dirichlet_constraint('right')
n_dofs = dQ.n_dofs(subforest_flag=mflag)
y = np.empty((n_dofs,n_samples))
for n in range(n_samples):
system.set_matrix(A[n])
system.set_rhs(b.copy())
system.solve_system()
y[:,n] = system.get_solution(as_function=False)[:,0]
y_fn = Nodal(dofhandler=dQ,subforest_flag=mflag,data=y)
if reference:
return y_fn, M, K
else:
return y_fn
def test_n_samples(self):
#
# Sampled Case
#
meshes = {1: Mesh1D(), 2: QuadMesh()}
elements = {1: QuadFE(1, 'Q2'), 2: QuadFE(2, 'Q2')}
# Use function to set data
fns = {
1: {
1: lambda x: 2 * x[:, 0]**2,
2: lambda x, y: 2 * x[:, 0] + 2 * y[:, 0]
},
2: {
1: lambda x: x[:, 0]**2 + x[:, 1],
2: lambda x, y: x[:, 0] * y[:, 0] + x[:, 1] * y[:, 1]
}
}
# n_samples = 2
parms = {1: {1: [{}, {}], 2: [{}, {}]}, 2: {1: [{}, {}], 2: [{}, {}]}}
for dim in [1, 2]:
mesh = meshes[dim]
element = elements[dim]
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
basis = Basis(dofhandler)
for n_variables in [1, 2]:
fn = fns[dim][n_variables]
parm = parms[dim][n_variables]
#
# Deterministic
#
f = Nodal(f=fn,
mesh=mesh,
basis=basis,
element=element,
dim=dim,
n_variables=n_variables)
self.assertEqual(f.n_samples(), 1)
#
# Sampled
#
f = Nodal(f=fn,
parameters=parm,
basis=basis,
mesh=mesh,
element=element,
dim=dim,
n_variables=n_variables)
self.assertEqual(f.n_samples(), 2)
def test_same_dofs(self):
#
# Construct nested mesh
#
mesh = QuadMesh()
mesh.record(0)
for dummy in range(2):
mesh.cells.refine()
#
# Define dofhandler
#
element = QuadFE(mesh.dim(), 'Q1')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
#
# Define basis functions
#
phi0 = Basis(dofhandler, 'u', subforest_flag=0)
phi0_x = Basis(dofhandler, 'ux', subforest_flag=0)
phi1 = Basis(dofhandler, 'u')
self.assertTrue(phi0.same_mesh(phi0_x))
self.assertFalse(phi0.same_mesh(phi1))
def test02a_sensitivity_gradient():
"""
Test whether the sensitivity and adjoint calculations give the same gradient
"""
# Mesh
mesh = Mesh1D(resolution=(100, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-10)
mesh.mark_region('right', lambda x: np.abs(1 - x) < 1e-10)
# Element
Q = QuadFE(mesh.dim(), 'Q2')
dh = DofHandler(mesh, Q)
dh.distribute_dofs()
n_dofs = dh.n_dofs()
phi = Basis(dh, 'u')
# Covariance
cov = Covariance(dh, name='gaussian', parameters={'l': 0.05})
cov.compute_eig_decomp()
lmd, V = cov.get_eig_decomp()
d = len(lmd)
# Coarse field (single sample)
d0 = 2
z0 = np.random.randn(d0, 1)
q0 = sample_q0(V, lmd, d0, z0)
q0_fn = Nodal(data=q0, basis=phi)
# State
J0, u0 = sample_qoi(q0, dh, return_state=True)
u0_fn = Nodal(data=u0, basis=phi)
# Compute gradient using sensitivity
dJs = np.zeros(n_dofs)
for i in range(n_dofs):
# Define perturbation
dq = np.zeros(n_dofs)
dq[i] = 1
dq_fn = Nodal(data=dq, basis=phi)
# Compute gradient using sensitivity
dJs[i] = dJdq_sen(q0_fn, u0_fn, dq_fn)
dJs_fn = Nodal(data=dJs, basis=phi)
plot = Plot()
plot.line(dJs_fn)
# Compute gradient using adjoint method
dJa = dJdq_adj(q0_fn, u0_fn)
dJa_fn = Nodal(data=dJa, basis=phi)
print(dJa)
plot.line(dJa_fn)
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))
def test02_variance():
"""
Compute the variance of J(q) for different mesh refinement levels
and compare with MC estimates.
"""
l_max = 8
for i_res in np.arange(2, l_max):
# Computational mesh
mesh = Mesh1D(resolution=(2**i_res, ))
# Element
element = QuadFE(mesh.dim(), 'DQ0')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Linear Functional
mesh.mark_region('integrate',
lambda x: x >= 0.75,
entity_type='cell',
strict_containment=False)
phi = Basis(dofhandler)
assembler = Assembler(Form(4, test=phi, flag='integrate'))
assembler.assemble()
L = assembler.get_vector()
# Define Gaussian random field
C = Covariance(dofhandler, name='gaussian', parameters={'l': 0.05})
C.compute_eig_decomp()
eta = GaussianField(dofhandler.n_dofs(), K=C)
eta.update_support()
n_samples = 100000
J_paths = L.dot(eta.sample(n_samples=n_samples))
var_mc = np.var(J_paths)
lmd, V = C.get_eig_decomp()
LV = L.dot(V)
var_an = LV.dot(np.diag(lmd).dot(LV.transpose()))
print(var_mc, var_an)
def test_constructor(self):
#
# Errors
#
# Nothing specified
self.assertRaises(Exception, Nodal)
# Nominal case
mesh = QuadMesh()
element = QuadFE(2, 'Q1')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
basis = Basis(dofhandler)
data = np.arange(0, 4)
f = Nodal(data=data, basis=basis, mesh=mesh, element=element)
self.assertEqual(f.dim(), 2)
self.assertTrue(np.allclose(f.data().ravel(), data))
# Now change the data -> Error
false_data = np.arange(0, 6)
self.assertRaises(
Exception, Nodal, **{
'data': false_data,
'mesh': mesh,
'element': element
})
# Now omit mesh or element
kwargs = {'data': data, 'mesh': mesh}
self.assertRaises(Exception, Nodal, **kwargs)
kwargs = {'data': data, 'element': element}
self.assertRaises(Exception, Nodal, **kwargs)
Q0 = QuadFE(1, 'DQ0') Q1 = QuadFE(1, 'Q1') # # Exact diffusion coefficient # qe = Function(qfn, 'explicit', dim=1) one = Function(1, 'constant') k1 = 1e-9 k2 = 1000 # # Basis functions # u = Basis(Q1, 'u') ux = Basis(Q1, 'ux') q = Basis(Q1, 'q') # # Forms # a_qe = Form(kernel=Kernel(qe), trial=ux, test=ux) a_one = Form(kernel=Kernel(one), trial=ux, test=ux) L = Form(kernel=Kernel(one), test=u) # # Problems # problems = [[a_qe,L], [a_one]]
mesh.mark_region('right', lambda x: np.abs(x - 2) < 1e-9)
#
# Finite element spaces
#
Q1 = QuadFE(mesh.dim(), 'Q1')
# Dofhandler for state
dh = DofHandler(mesh, Q1)
dh.distribute_dofs()
m = dh.n_dofs()
dh.set_dof_vertices()
x = dh.get_dof_vertices()
# Basis functions
phi = Basis(dh, 'v')
phi_x = Basis(dh, 'vx')
state = LS(phi)
state.add_dirichlet_constraint('left', 1)
state.add_dirichlet_constraint('right', 0)
state.set_constraint_relation()
adjoint = LS(phi)
adjoint.add_dirichlet_constraint('left', 0)
adjoint.add_dirichlet_constraint('right', 0)
adjoint.set_constraint_relation()
# =============================================================================
# System Parameters
# =============================================================================
errors[resolution] = {}
for eps in [1, 1e-3, 1e-6]:
errors[resolution][eps] = {}
for etype in ['Q1', 'Q2']:
#
# Define element
#
element = QuadFE(2, etype)
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
#
# Define Basis Functions
#
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
uy = Basis(dofhandler, 'uy')
#
# Define weak form
#
a_diff_x = Form(eps, trial=ux, test=ux)
a_diff_y = Form(eps, trial=uy, test=uy)
a_adv_x = Form(vx, trial=ux, test=u)
a_adv_y = Form(vy, trial=uy, test=u)
b = Form(f, test=u)
problem = [a_diff_x, a_diff_y, a_adv_x, a_adv_y, b]
#
# Define Mesh
#
mesh = QuadMesh(resolution=(2,1))
mesh.cells.get_child(1).mark('1')
mesh.cells.refine(refinement_flag='1')
#
# Define element
#
Q1 = QuadFE(2,'Q1')
#
# Basis Functions
#
u = Basis(Q1, 'u')
ux = Basis(Q1, 'ux')
uy = Basis(Q1, 'uy')
rule = GaussRule(9, Q1)
ue = Function(lambda x,dummy: x, 'nodal', mesh=mesh, element=Q1)
zero = Function(0, 'constant')
ax = Form(trial=ux, test=ux)
ay = Form(trial=uy, test=uy)
L = Form(kernel=Kernel(zero), test=u)
assembler = Assembler([ax,ay,L], mesh)
for dofhandler in assembler.dofhandlers.values():
dofhandler.set_hanging_nodes()
mesh.mark_region('perimeter', pf, entity_type='half_edge')
# Get rid of neighbors of half-edges on slit
for he in mesh.half_edges.get_leaves('slit'):
if he.unit_normal()[0] < 0:
he.mark('B')
mesh.tear_region('B')
# =============================================================================
# Functions
# =============================================================================
Q1 = QuadFE(2, 'Q1')
dofhandler = DofHandler(mesh, Q1)
dofhandler.distribute_dofs()
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
uy = Basis(dofhandler, 'uy')
epsilon = 1e-6
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)
]
f_int_region = lambda x: x >= x_min and x <= x_max
f_cnd_region = lambda x: x >= xx_min and x <= xx_max
mesh.mark_region('integration', f_int_region, entity_type='cell')
mesh.mark_region('condition', f_cnd_region, entity_type='cell')
#
# Finite Elements
#
# Piecewise Constant
Q0 = QuadFE(mesh.dim(), 'DQ0')
dQ0 = DofHandler(mesh, Q0)
dQ0.distribute_dofs()
phi_0 = Basis(dQ0)
# Piecewise Linear
Q1 = QuadFE(mesh.dim(), 'Q1')
dQ1 = DofHandler(mesh, Q1)
dQ1.distribute_dofs()
phi_1 = Basis(dQ1)
# -----------------------------------------------------------------------------
# Stochastic Approximation
# -----------------------------------------------------------------------------
#
# Random Field
#
# Covariance kernel
K = Covariance(dQ1, name='gaussian', parameters={'sgm': 1, 'l': 0.1})
def test09_1d_inverse(self):
"""
Compute the inverse of a matrix and apply it to a vector/matrix.
"""
#
# Mesh
#
mesh = Mesh1D(resolution=(1, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-9, on_boundary=True)
mesh.mark_region('right',
lambda x: np.abs(1 - x) < 1e-9,
on_boundary=True)
#
# Elements
#
Q3 = QuadFE(1, 'Q3')
dofhandler = DofHandler(mesh, Q3)
dofhandler.distribute_dofs()
#
# Basis
#
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
#
# Define sampled right hand side and exact solution
#
xv = dofhandler.get_dof_vertices()
n_points = dofhandler.n_dofs()
n_samples = 6
a = np.arange(n_samples)
ffn = lambda x, a: a * x
ufn = lambda x, a: a / 6 * (x - x**3) + x
fdata = np.zeros((n_points, n_samples))
udata = np.zeros((n_points, n_samples))
for i in range(n_samples):
fdata[:, i] = ffn(xv, a[i]).ravel()
udata[:, i] = ufn(xv, a[i]).ravel()
# Define sampled function
fn = Nodal(data=fdata, basis=u)
ue = Nodal(data=udata, basis=u)
#
# Forms
#
one = Constant(1)
a = Form(Kernel(one), test=ux, trial=ux)
L = Form(Kernel(fn), test=u)
problem = [[a], [L]]
#
# Assembler
#
assembler = Assembler(problem, mesh)
assembler.assemble()
A = assembler.get_matrix()
b = assembler.get_vector(i_problem=1)
#
# Linear System
#
system = LinearSystem(u, A=A)
# Set constraints
system.add_dirichlet_constraint('left', 0)
system.add_dirichlet_constraint('right', 1)
system.solve_system(b)
# Extract finite element solution
ua = system.get_solution(as_function=True)
system2 = LinearSystem(u, A=A, b=b)
# Set constraints
system2.add_dirichlet_constraint('left', 0)
system2.add_dirichlet_constraint('right', 1)
system2.solve_system()
u2 = system2.get_solution(as_function=True)
# Check that the solution is close
self.assertTrue(np.allclose(ue.data()[:, 0], ua.data()[:, 0]))
self.assertTrue(np.allclose(ue.data()[:, [0]], u2.data()))
def test08_1d_sampled_rhs(self):
#
# Mesh
#
mesh = Mesh1D(resolution=(1, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-9, on_boundary=True)
mesh.mark_region('right',
lambda x: np.abs(1 - x) < 1e-9,
on_boundary=True)
#
# Elements
#
Q3 = QuadFE(1, 'Q3')
dofhandler = DofHandler(mesh, Q3)
dofhandler.distribute_dofs()
#
# Basis
#
v = Basis(dofhandler, 'u')
vx = Basis(dofhandler, 'ux')
#
# Define sampled right hand side and exact solution
#
xv = dofhandler.get_dof_vertices()
n_points = dofhandler.n_dofs()
n_samples = 6
a = np.arange(n_samples)
f = lambda x, a: a * x
u = lambda x, a: a / 6 * (x - x**3) + x
fdata = np.zeros((n_points, n_samples))
udata = np.zeros((n_points, n_samples))
for i in range(n_samples):
fdata[:, i] = f(xv, a[i]).ravel()
udata[:, i] = u(xv, a[i]).ravel()
# Define sampled function
fn = Nodal(data=fdata, basis=v)
ue = Nodal(data=udata, basis=v)
#
# Forms
#
one = Constant(1)
a = Form(Kernel(one), test=vx, trial=vx)
L = Form(Kernel(fn), test=v)
problem = [a, L]
#
# Assembler
#
assembler = Assembler(problem, mesh)
assembler.assemble()
A = assembler.get_matrix()
b = assembler.get_vector()
#
# Linear System
#
system = LinearSystem(v, A=A, b=b)
# Set constraints
system.add_dirichlet_constraint('left', 0)
system.add_dirichlet_constraint('right', 1)
#system.set_constraint_relation()
#system.incorporate_constraints()
# Solve and resolve constraints
system.solve_system()
#system.resolve_constraints()
# Extract finite element solution
ua = system.get_solution(as_function=True)
# Check that the solution is close
print(ue.data()[:, [0]])
print(ua.data())
self.assertTrue(np.allclose(ue.data()[:, [0]], ua.data()))
def test05_2d_dirichlet(self):
"""
Two dimensional Dirichlet problem with hanging nodes
"""
#
# Define mesh
#
mesh = QuadMesh(resolution=(1, 2))
mesh.cells.get_child(1).mark(1)
mesh.cells.refine(refinement_flag=1)
mesh.cells.refine()
#
# 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']:
#
# Element
#
element = QuadFE(2, etype)
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
#
# Basis
#
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
uy = Basis(dofhandler, 'uy')
#
# Construct forms
#
ue = Nodal(f=lambda x: x[:, 0], basis=u)
ax = Form(kernel=Kernel(Constant(1)), trial=ux, test=ux)
ay = Form(kernel=Kernel(Constant(1)), trial=uy, test=uy)
L = Form(kernel=Kernel(Constant(0)), test=u)
problem = [ax, ay, L]
#
# Assemble
#
assembler = Assembler(problem, mesh)
assembler.assemble()
#
# Get system matrices
#
A = assembler.get_matrix()
b = assembler.get_vector()
#
# Linear System
#
system = LinearSystem(u, A=A, b=b)
#
# Constraints
#
# Add dirichlet conditions
system.add_dirichlet_constraint('left', ue)
system.add_dirichlet_constraint('right', ue)
#
# Solve
#
system.solve_system()
#system.resolve_constraints()
#
# Check solution
#
ua = system.get_solution(as_function=True)
self.assertTrue(np.allclose(ua.data(), ue.data()))
def test01_1d_dirichlet_linear(self):
"""
Solve one dimensional boundary value problem with dirichlet
conditions on left and right
"""
#
# Define mesh
#
mesh = Mesh1D(resolution=(10, ))
for etype in ['Q1', 'Q2', 'Q3']:
element = QuadFE(1, etype)
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
phi = Basis(dofhandler)
#
# Exact solution
#
ue = Nodal(f=lambda x: x, basis=phi)
#
# Define Basis functions
#
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
#
# Define bilinear form
#
one = Constant(1)
zero = Constant(0)
a = Form(kernel=Kernel(one), trial=ux, test=ux)
L = Form(kernel=Kernel(zero), test=u)
problem = [a, L]
#
# Assemble
#
assembler = Assembler(problem, mesh)
assembler.assemble()
#
# Form linear system
#
A = assembler.get_matrix()
b = assembler.get_vector()
system = LinearSystem(u, A=A, b=b)
#
# Dirichlet conditions
#
# Boundary functions
bm_left = lambda x: np.abs(x) < 1e-9
bm_rght = lambda x: np.abs(x - 1) < 1e-9
# Mark boundary regions
mesh.mark_region('left', bm_left, on_boundary=True)
mesh.mark_region('right', bm_rght, on_boundary=True)
# Add Dirichlet constraints
system.add_dirichlet_constraint('left', ue)
system.add_dirichlet_constraint('right', ue)
#
# Solve system
#
#system.solve_system()
system.solve_system()
#
# Get solution
#
#ua = system.get_solution(as_function=True)
uaa = system.get_solution(as_function=True)
#uaa = uaa.data().ravel()
# Compare with exact solution
#self.assertTrue(np.allclose(ua.data(), ue.data()))
self.assertTrue(np.allclose(uaa.data(), ue.data()))
def test04_1d_periodic(self):
#
# Dirichlet Problem on a Periodic Mesh
#
# Define mesh, element
mesh = Mesh1D(resolution=(100, ), periodic=True)
element = QuadFE(1, 'Q3')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Basis functions
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
# Exact solution
ue = Nodal(f=lambda x: np.sin(2 * np.pi * x), basis=u)
#
# Mark dirichlet regions
#
bnd_left = lambda x: np.abs(x) < 1e-9
mesh.mark_region('left', bnd_left, entity_type='vertex')
#
# Set up forms
#
# Bilinear form
a = Form(kernel=Kernel(Constant(1)), trial=ux, test=ux)
# Linear form
f = Explicit(lambda x: 4 * np.pi**2 * np.sin(2 * np.pi * x), dim=1)
L = Form(kernel=Kernel(f), test=u)
#
# Assemble
#
problem = [a, L]
assembler = Assembler(problem, mesh)
assembler.assemble()
A = assembler.get_matrix()
b = assembler.get_vector()
#
# Linear System
#
system = LinearSystem(u, A=A, b=b)
# Add dirichlet constraint
system.add_dirichlet_constraint('left', 0, on_boundary=False)
# Assemble constraints
#system.set_constraint_relation()
#system.incorporate_constraints()
system.solve_system()
#system.resolve_constraints()
# Compare with interpolant of exact solution
ua = system.get_solution(as_function=True)
#plot = Plot(2)
#plot.line(ua)
#plot.line(ue)
self.assertTrue(np.allclose(ua.data(), ue.data()))
