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 test_eval(self):
#
# 1D
#
# Define Kernel function
def k_fn(x,y,c = 2):
return x*y + c
k = Explicit(f=lambda x,y,c: x*y+c, parameters={'c':1},
n_variables=2, dim=1)
# Construct kernel, specifying parameter c
kernel = Kernel(k)
# Evaluation points
x = np.ones((11,1))
y = np.linspace(0,1,11)[:,None]
# Check accuracy
self.assertTrue(np.allclose(kernel.eval((x,y)), x*y+1))
# Define kernel with default parameters
k.set_parameters({'c':2})
kernel = Kernel(k)
# Check accuracy
self.assertTrue(np.allclose(kernel.eval((x,y)), x*y+2))
def test_constructor(self):
"""
Test initialization
"""
f = Explicit(lambda x, a: a * x**2,
parameters={'a': 1},
dim=1,
n_variables=1)
f.eval(1)
self.assertAlmostEqual(f.eval(1), 1)
#
# Errors
#
self.assertRaises(Exception, Explicit)
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 test_add_rules(self):
# Define explicit function
rule = lambda x, a: a * x**2
parameters = [{'a': 1}, {'a': 2}]
f = Explicit(rule, parameters=parameters, dim=1)
# Raises an error attempt to add multiple functions
self.assertRaises(Exception, f.add_rule, *(lambda x: x, ),
**{'parameters': [{}, {}]})
# Add a single rule and evaluate at a point
f.add_rule(lambda x: x)
x = 2
vals = [4, 8, 2]
for i in range(3):
#(*(x,),**{'a':1}))
self.assertEqual(f.eval(x)[0, i], vals[i])
def test_n_samples(self):
#
# 1D
#
# Define mesh
mesh = Mesh1D(resolution=(10,))
# Define function
f = Explicit([lambda x: x, lambda x: -2+2*x**2], dim=1)
n_samples = f.n_samples()
k = Kernel(f)
n_points = 101
x0, x1 = mesh.bounding_box()
x = np.linspace(x0,x1,n_points)
self.assertEqual(k.eval(x).shape, (n_points, n_samples))
self.assertTrue(np.allclose(k.eval(x)[:,0],x))
self.assertTrue(np.allclose(k.eval(x)[:,1], -2+2*x**2))
def test_set_rules(self):
#
# Errors
#
# Number of parameter dicts differs from number of rules
flist = [lambda x, a: a * x**2, lambda x: x]
plist = [{}, {}, {}]
self.assertRaises(Exception, Explicit, *(flist, ), **{
'parameters': plist,
'dim': 1
})
# Specify that function is symmetric, but n_variables=1
f = lambda x: x
self.assertRaises(Exception, Explicit, *(f, ), **{'symmetric': True})
#
# Expected
#
# Only one set of parameters multiple functions
f = Explicit(flist, parameters={}, dim=1)
self.assertEqual(len(f.parameters()), 2)
# Only one function multiple parameters
rule = lambda x, a: a * x**2
parameters = [{'a': 1}, {'a': 2}]
f = Explicit(rule, parameters=parameters, dim=1)
self.assertEqual(len(f.rules()), 2)
def test_set_parameters(self):
# Define explicit function
rule = lambda x, a: a * x[:, 0] - x[:, 1]
parms = [{'a': 1}, {'a': 2}]
f = Explicit(rule, parameters=parms, dim=2)
x = (1, 2)
self.assertTrue(np.allclose(f.eval(x), np.array([-1, 0])))
# Modify parameter
f.set_parameters({'a': 2}, pos=0)
self.assertTrue(np.allclose(f.eval(x), np.array([0, 0])))
def test_eval_interpolation(self):
#
# 1D
#
# Mesh
mesh = Mesh1D(resolution=(3, ))
# Finite element space
etype = 'Q1'
element = QuadFE(1, etype)
# Dofhandler
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Basis functions
phi = Basis(dofhandler, 'u')
# Symmetric kernel function
kfns = {
'symmetric': lambda x, y: x * y,
'non_symmetric': lambda x, y: x - y
}
vals = {
'symmetric':
np.array([0, 1 / 9 * (1 - 8 / 27)]),
'non_symmetric':
np.array([1 / 3 * (-1 + 8 / 27), 1 / 6 * 5 / 9 - 1 / 3 * 19 / 27])
}
for ktype in ['symmetric', 'non_symmetric']:
# Get kernel function
kfn = kfns[ktype]
# Define integral kernel
kernel = Kernel(Explicit(kfn, dim=1, n_variables=2))
# Define Bilinear Form
form = IIForm(kernel, trial=phi, test=phi)
#
# Compute inputs required for evaluating form_loc
#
# Assembler
assembler = Assembler(form, mesh)
# Cells
cj = mesh.cells.get_child(2)
ci = mesh.cells.get_child(0)
# Gauss nodes and weights on cell
xj_g, wj_g, phij, dofsj = assembler.shape_eval(cj)
#
# Evaluate form
#
form_loc = form.eval(cj, xj_g, wj_g, phij, dofsj)
#
# Define functions
#
u = Nodal(lambda x: x, basis=phi)
#
# Get local node values
#
# Degrees of freedom
cj_dofs = phi.dofs(cj)
ci_dofs = phi.dofs(ci)
uj = u.data()[np.array(cj_dofs)]
# Evaluate Ici Icj k(x,y) y dy (1-x)dx
fa = form_loc[ci_dofs].dot(uj)
fe = vals[ktype][:, None]
self.assertTrue(np.allclose(fa, fe))
def test_subsample_deterministic(self):
"""
When evaluating a deterministic function while specifying a subsample,
n_subsample copies of the function output should be returned.
"""
#
# Deterministic functions
#
# Functions
fns = {
1: {
1: lambda x: x[:, 0]**2,
2: lambda x, y: x[:, 0] + y[:, 0]
},
2: {
1: lambda x: x[:, 0]**2 + x[:, 1]**2,
2: lambda x, y: x[:, 0] * y[:, 0] + x[:, 1] * y[:, 1]
}
}
# Singletons
x = {1: {1: 2, 2: (3, 4)}, 2: {1: (1, 2), 2: ((1, 2), (3, 4))}}
xv = {
1: {
1: [(2, ), (2, )],
2: ([(3, ), (3, )], [(4, ), (4, )])
},
2: {
1: [(1, 2), (1, 2)],
2: ([(1, 2), (1, 2)], [(3, 4), (3, 4)])
}
}
vals = {1: {1: 4, 2: 7}, 2: {1: 5, 2: 11}}
subsample = np.array([2, 3], dtype=np.int)
for dim in [1, 2]:
#
# Iterate over dimension
#
# DofHandler
if dim == 1:
mesh = Mesh1D(box=[0, 5], resolution=(1, ))
elif dim == 2:
mesh = QuadMesh(box=[0, 5, 0, 5])
element = QuadFE(dim, 'Q2')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
basis = Basis(dofhandler)
for n_variables in [1, 2]:
#
# Iterate over number of variables
#
#
# Explicit
#
f = fns[dim][n_variables]
# Explicit
fe = Explicit(f, n_variables=n_variables, dim=dim, \
subsample=subsample)
# Nodal
fn = Nodal(f, n_variables=n_variables, basis=basis, dim=dim, \
dofhandler=dofhandler, subsample=subsample)
# Constant
fc = Constant(1, n_variables=n_variables, \
subsample=subsample)
# Singleton input
xn = x[dim][n_variables]
# Explicit
self.assertEqual(fe.eval(xn).shape[1], len(subsample))
self.assertEqual(fe.eval(xn)[0, 0], vals[dim][n_variables])
self.assertEqual(fe.eval(xn)[0, 1], vals[dim][n_variables])
# Nodal
self.assertEqual(fn.eval(xn).shape[1], len(subsample))
self.assertAlmostEqual(
fn.eval(xn)[0, 0], vals[dim][n_variables])
self.assertAlmostEqual(
fn.eval(xn)[0, 1], vals[dim][n_variables])
# Constant
self.assertEqual(fc.eval(xn).shape[1], len(subsample))
self.assertAlmostEqual(fc.eval(xn)[0, 0], 1)
self.assertAlmostEqual(fc.eval(xn)[0, 1], 1)
# Vector input
xn = xv[dim][n_variables]
n_points = 2
# Explicit
self.assertEqual(fe.eval(xn).shape, (2, 2))
for i in range(fe.n_subsample()):
for j in range(n_points):
self.assertEqual(
fe.eval(xn)[i][j], vals[dim][n_variables])
# Nodal
self.assertEqual(fn.eval(xn).shape, (2, 2))
for i in range(fe.n_subsample()):
for j in range(n_points):
self.assertAlmostEqual(
fn.eval(xn)[i][j], vals[dim][n_variables])
# Constant
self.assertEqual(fc.eval(xn).shape, (2, 2))
for i in range(fe.n_subsample()):
for j in range(n_points):
self.assertEqual(fc.eval(xn)[i][j], 1)
def test_eval(self):
#
# Evaluate single univariate/bivariate functions in 1 or 2 dimensions
#
fns = {
1: {
1: lambda x: x**2,
2: lambda x, y: x + y
},
2: {
1: lambda x: x[:, 0]**2 + x[:, 1]**2,
2: lambda x, y: x[:, 0] * y[:, 0] + x[:, 1] * y[:, 1]
}
}
# Singletons
x = {1: {1: 2, 2: (3, 4)}, 2: {1: (1, 2), 2: ((1, 2), (3, 4))}}
vals = {1: {1: 4, 2: 7}, 2: {1: 5, 2: 11}}
for dim in [1, 2]:
#
# Iterate over dimension
#
for n_variables in [1, 2]:
#
# Iterate over number of variables
#
fn = fns[dim][n_variables]
f = Explicit(fn, n_variables=n_variables, dim=dim)
xn = x[dim][n_variables]
self.assertEqual(f.eval(xn), vals[dim][n_variables])
#
# Evaluate sampled functions
#
fns = {
1: {
1: lambda x, a: a * x**2,
2: lambda x, y, a: a * (x + y)
},
2: {
1: lambda x, a: a * (x[:, 0]**2 + x[:, 1]**2),
2: lambda x, y, a: a * (x[:, 0] * y[:, 0] + x[:, 1] * y[:, 1])
}
}
pars = [{'a': 1}, {'a': 2}]
#
# Singletons
#
x = {1: {1: 2, 2: (3, 4)}, 2: {1: (1, 2), 2: ((1, 2), (3, 4))}}
vals = {1: {1: 4, 2: 7}, 2: {1: 5, 2: 11}}
for dim in [1, 2]:
#
# Iterate over dimension
#
for n_variables in [1, 2]:
#
# Iterate over number of variables
#
fn = fns[dim][n_variables]
f = Explicit(fn,
parameters=pars,
n_variables=n_variables,
dim=dim)
xn = x[dim][n_variables]
self.assertEqual(f.eval(xn)[0][0], vals[dim][n_variables])
self.assertEqual(f.eval(xn)[0][1], 2 * vals[dim][n_variables])
#
# 2 points
#
n_points = 2
x = {
1: {
1: [(2, ), (2, )],
2: ([(3, ), (3, )], [(4, ), (4, )])
},
2: {
1: [(1, 2), (1, 2)],
2: ([(1, 2), (1, 2)], [(3, 4), (3, 4)])
}
}
for dim in [1, 2]:
#
# Iterate over dimension
#
for n_variables in [1, 2]:
#
# Iterate over number of variables
#
fn = fns[dim][n_variables]
f = Explicit(fn,
parameters=pars,
n_variables=n_variables,
dim=dim)
xn = x[dim][n_variables]
self.assertEqual(f.eval(xn).shape[0], n_points)
self.assertEqual(f.eval(xn).shape[1], f.n_samples())
for i in range(f.n_samples()):
for j in range(2):
val = pars[i]['a'] * vals[dim][n_variables]
self.assertEqual(f.eval(xn)[j, i], val)
def test_integrals_2d(self):
"""
Test Assembly of some 2D systems
"""
mesh = QuadMesh(box=[1, 2, 1, 2], resolution=(2, 2))
mesh.cells.get_leaves()[0].mark(0)
mesh.cells.refine(refinement_flag=0)
# Kernel
kernel = Kernel(Explicit(f=lambda x: x[:, 0] * x[:, 1], dim=2))
problem = Form(kernel)
assembler = Assembler(problem, mesh=mesh)
assembler.assemble()
self.assertAlmostEqual(assembler.get_scalar(), 9 / 4)
#
# Linear forms (x,x) and (x,x') over [1,2]^2 = 7/3, 3/2
#
# Elements
Q1 = QuadFE(mesh.dim(), 'Q1')
Q2 = QuadFE(mesh.dim(), 'Q2')
Q3 = QuadFE(mesh.dim(), 'Q3')
# Dofhandlers
dQ1 = DofHandler(mesh, Q1)
dQ2 = DofHandler(mesh, Q2)
dQ3 = DofHandler(mesh, Q3)
# Distribute dofs
[d.distribute_dofs() for d in [dQ1, dQ2, dQ3]]
for dQ in [dQ1, dQ2, dQ3]:
# Basis
phi = Basis(dQ, 'u')
phi_x = Basis(dQ, 'ux')
# Kernel function
xfn = Nodal(f=lambda x: x[:, 0], basis=phi)
yfn = Nodal(f=lambda x: x[:, 1], basis=phi)
# Kernel
kernel = Kernel(xfn)
# Form
problem = [[Form(kernel, test=phi)], [Form(kernel, test=phi_x)]]
# Assembly
assembler = Assembler(problem, mesh)
assembler.assemble()
# Check b^Tx = (x,y)
b0 = assembler.get_vector(0)
self.assertAlmostEqual(np.sum(b0 * yfn.data()[:, 0]), 9 / 4)
b1 = assembler.get_vector(1)
self.assertAlmostEqual(np.sum(b1 * xfn.data()[:, 0]), 3 / 2)
self.assertAlmostEqual(np.sum(b1 * yfn.data()[:, 0]), 0)
#
# Bilinear forms
#
# Compute (1,x,y) = 9/4, or (xy, 1, 1) = 9/4
for dQ in [dQ1, dQ2, dQ3]:
# Basis
phi = Basis(dQ, 'u')
phi_x = Basis(dQ, 'ux')
phi_y = Basis(dQ, 'uy')
# Kernel function
xyfn = Explicit(f=lambda x: x[:, 0] * x[:, 1], dim=2)
xfn = Nodal(f=lambda x: x[:, 0], basis=phi)
yfn = Nodal(f=lambda x: x[:, 1], basis=phi)
# Form
problems = [[Form(1, test=phi, trial=phi)],
[Form(Kernel(xfn), test=phi, trial=phi_x)],
[Form(Kernel(xyfn), test=phi_y, trial=phi_x)]]
# Assemble
assembler = Assembler(problems, mesh)
assembler.assemble()
x = xfn.data()[:, 0]
y = yfn.data()[:, 0]
for i_problem in range(3):
A = assembler.get_matrix(i_problem)
self.assertAlmostEqual(y.T.dot(A.dot(x)), 9 / 4)
from mesh import QuadMesh, Mesh1D
from plot import Plot
from fem import QuadFE, DofHandler
from function import Explicit
import numpy as np
plot = Plot()
mesh = Mesh1D()
Q0 = QuadFE(1, 'DQ0')
dh0 = DofHandler(mesh, Q0)
n_levels = 10
for l in range(n_levels):
mesh.cells.refine(new_label=l)
dh0.distribute_dofs(subforest_flag=l)
f = Explicit(lambda x: np.abs(x - 0.5), dim=1)
fQ = f.interpolant(dh0, subforest_flag=3)
plot.line(fQ, mesh)
plot.mesh(mesh, dofhandler=dh0, subforest_flag=0)
mesh = QuadMesh(resolution=(10, 10))
plot.mesh(mesh)
dofhandler = DofHandler(mesh, element)
comment.tic('distribute dofs')
dofhandler.distribute_dofs()
comment.toc()
print('number of dofs:', dofhandler.n_dofs())
# Basis functions
p_u = Basis(dofhandler, 'u')
p_ux = Basis(dofhandler, 'ux')
p_uy = Basis(dofhandler, 'uy')
pp_u = Basis(dofhandler, 'u')
f = lambda x: x[:, 0] * (1 - x[:, 0]) * x[:, 1] * (1 - x[:, 1])
qq = Explicit(f=f, dim=2)
fNodal = Nodal(f=f, basis=p_u)
qk = Kernel(f=[qq], F=lambda qq, mu=3: 3 * qq)
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
# =============================================================================
# Define problem
flow_problem = [
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)
]
print('assembling', end=' ')
assembler = Assembler(problem, mesh)
assembler.assemble()
print('done')
def test_integrals_1d(self):
"""
Test system assembly
"""
#
# Constant form
#
# Mesh
mesh = Mesh1D(box=[1, 2], resolution=(1, ))
# Kernel
kernel = Kernel(Explicit(f=lambda x: x[:, 0], dim=1))
problem = Form(kernel)
assembler = Assembler(problem, mesh=mesh)
assembler.assemble()
self.assertAlmostEqual(assembler.get_scalar(), 3 / 2)
#
# Linear forms (x,x) and (x,x') over [1,2] = 7/3, 3/2
#
# Elements
Q1 = QuadFE(mesh.dim(), 'Q1')
Q2 = QuadFE(mesh.dim(), 'Q2')
Q3 = QuadFE(mesh.dim(), 'Q3')
# Dofhandlers
dQ1 = DofHandler(mesh, Q1)
dQ2 = DofHandler(mesh, Q2)
dQ3 = DofHandler(mesh, Q3)
# Distribute dofs
[d.distribute_dofs() for d in [dQ1, dQ2, dQ3]]
for dQ in [dQ1, dQ2, dQ3]:
# Basis
phi = Basis(dQ, 'u')
phi_x = Basis(dQ, 'ux')
# Kernel function
xfn = Nodal(f=lambda x: x[:, 0], basis=phi)
# Kernel
kernel = Kernel(xfn)
# Form
problem = [[Form(kernel, test=phi)], [Form(kernel, test=phi_x)]]
# Assembly
assembler = Assembler(problem, mesh)
assembler.assemble()
# Check b^Tx = (x,x)
b0 = assembler.get_vector(0)
self.assertAlmostEqual(np.sum(b0 * xfn.data()[:, 0]), 7 / 3)
b1 = assembler.get_vector(1)
self.assertAlmostEqual(np.sum(b1 * xfn.data()[:, 0]), 3 / 2)
#
# Bilinear forms
#
# Compute (1,x,x) = 7/3, or (x^2, 1, 1) = 7/3
for dQ in [dQ1, dQ2, dQ3]:
# Basis
phi = Basis(dQ, 'u')
phi_x = Basis(dQ, 'ux')
# Kernel function
x2fn = Explicit(f=lambda x: x[:, 0]**2, dim=1)
xfn = Nodal(f=lambda x: x[:, 0], basis=phi)
# Form
problems = [[Form(1, test=phi, trial=phi)],
[Form(Kernel(xfn), test=phi, trial=phi_x)],
[Form(Kernel(x2fn), test=phi_x, trial=phi_x)]]
# Assemble
assembler = Assembler(problems, mesh)
assembler.assemble()
x = xfn.data()[:, 0]
for i_problem in range(3):
A = assembler.get_matrix(i_problem)
self.assertAlmostEqual(x.T.dot(A.dot(x)), 7 / 3)
# ======================================================================
# Test 1: Assemble simple bilinear form (u,v) on Mesh1D
# ======================================================================
# Mesh
mesh = Mesh1D(resolution=(1, ))
Q1 = QuadFE(mesh.dim(), 'Q1')
Q2 = QuadFE(mesh.dim(), 'Q2')
Q3 = QuadFE(mesh.dim(), 'Q3')
# Test and trial functions
dhQ1 = DofHandler(mesh, QuadFE(1, 'Q1'))
dhQ1.distribute_dofs()
u = Basis(dhQ1, 'u')
v = Basis(dhQ1, 'v')
# Form
form = Form(trial=u, test=v)
# Define system
system = Assembler(form, mesh)
# Get local information
cell = mesh.cells.get_child(0)
si = system.shape_info(cell)
# Compute local Gauss nodes
xg, wg, phi, dofs = system.shape_eval(cell)
self.assertTrue(cell in xg)
self.assertTrue(cell in wg)
# Compute local shape functions
self.assertTrue(cell in phi)
self.assertTrue(u in phi[cell])
self.assertTrue(v in phi[cell])
self.assertTrue(u in dofs[cell])
# Assemble system
system.assemble()
# Extract system bilinear form
A = system.get_matrix()
# Use bilinear form to integrate x^2 over [0,1]
f = Nodal(lambda x: x, basis=u)
fv = f.data()[:, 0]
self.assertAlmostEqual(np.sum(fv * A.dot(fv)), 1 / 3)
# ======================================================================
# Test 3: Constant form (x^2,.,.) over 1D mesh
# ======================================================================
# Mesh
mesh = Mesh1D(resolution=(10, ))
# Nodal kernel function
Q2 = QuadFE(1, 'Q2')
dhQ2 = DofHandler(mesh, Q2)
dhQ2.distribute_dofs()
phiQ2 = Basis(dhQ2)
f = Nodal(lambda x: x**2, basis=phiQ2)
kernel = Kernel(f=f)
# Form
form = Form(kernel=kernel)
# Generate and assemble the system
system = Assembler(form, mesh)
system.assemble()
# Check
self.assertAlmostEqual(system.get_scalar(), 1 / 3)
# =====================================================================
# Test 4: Periodic Mesh
# =====================================================================
#
# TODO: NO checks yet
#
mesh = Mesh1D(resolution=(2, ), periodic=True)
#
Q1 = QuadFE(1, 'Q1')
dhQ1 = DofHandler(mesh, Q1)
dhQ1.distribute_dofs()
u = Basis(dhQ1, 'u')
form = Form(trial=u, test=u)
system = Assembler(form, mesh)
system.assemble()
# =====================================================================
# Test 5: Assemble simple sampled form
# ======================================================================
mesh = Mesh1D(resolution=(3, ))
Q1 = QuadFE(1, 'Q1')
dofhandler = DofHandler(mesh, Q1)
dofhandler.distribute_dofs()
phi = Basis(dofhandler)
xv = dofhandler.get_dof_vertices()
n_points = dofhandler.n_dofs()
n_samples = 6
a = np.arange(n_samples)
f = lambda x, a: a * x
fdata = np.zeros((n_points, n_samples))
for i in range(n_samples):
fdata[:, i] = f(xv, a[i]).ravel()
# Define sampled function
fn = Nodal(data=fdata, basis=phi)
kernel = Kernel(fn)
#
# Integrate I[0,1] ax^2 dx by using the linear form (ax,x)
#
v = Basis(dofhandler, 'v')
form = Form(kernel=kernel, test=v)
system = Assembler(form, mesh)
system.assemble()
one = np.ones(n_points)
for i in range(n_samples):
b = system.get_vector(i_sample=i)
self.assertAlmostEqual(one.dot(b), 0.5 * a[i])
#
# Integrate I[0,1] ax^4 dx using bilinear form (ax, x^2, x)
#
Q2 = QuadFE(1, 'Q2')
dhQ2 = DofHandler(mesh, Q2)
dhQ2.distribute_dofs()
u = Basis(dhQ2, 'u')
# Define form
form = Form(kernel=kernel, test=v, trial=u)
# Define and assemble system
system = Assembler(form, mesh)
system.assemble()
# Express x^2 in terms of trial function basis
dhQ2.distribute_dofs()
xvQ2 = dhQ2.get_dof_vertices()
xv_squared = xvQ2**2
for i in range(n_samples):
#
# Iterate over samples
#
# Form sparse matrix
A = system.get_matrix(i_sample=i)
# Evaluate the integral
I = np.sum(xv * A.dot(xv_squared))
# Compare with expected result
self.assertAlmostEqual(I, 0.2 * a[i])
# =====================================================================
# Test 6: Working with submeshes
# =====================================================================
mesh = Mesh1D(resolution=(2, ))
def test_eval_projection(self):
"""
Test validity of the local projection-based kernel.
Choose
u, v in V
k(x,y) in VxV (symmetric/non-symm)
cell01, cell02
Compare
v^T*Kloc*u ?= Icell01 Icell02 k(x,y)u(y)dy dx
"""
#
# 1D
#
# Mesh
mesh = Mesh1D(resolution=(3, ))
# Finite element space
etype = 'Q1'
element = QuadFE(1, etype)
# Dofhandler
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Basis functions
phi = Basis(dofhandler, 'u')
# Symmetric kernel function
kfns = {
'symmetric': lambda x, y: x * y,
'non_symmetric': lambda x, y: x - y
}
vals = {
'symmetric':
(1 / 2 * 1 / 9 - 1 / 3 * 1 / 27) * (1 / 3 - 1 / 3 * 8 / 27),
'non_symmetric': (1 / 18 - 1 / 3 * 1 / 27) * (1 / 2 - 2 / 9) +
(1 / 18 - 1 / 3) * (1 / 3 - 1 / 3 * 8 / 27)
}
for ktype in ['symmetric', 'non_symmetric']:
# Get kernel function
kfn = kfns[ktype]
# Define integral kernel
kernel = Kernel(Explicit(kfn, dim=1, n_variables=2))
# Define Bilinear Form
form = IPForm(kernel, trial=phi, test=phi)
#
# Compute inputs required for evaluating form_loc
#
# Assembler
assembler = Assembler(form, mesh)
# Cells
ci = mesh.cells.get_child(0)
cj = mesh.cells.get_child(2)
# Shape function info on cells
ci_sinfo = assembler.shape_info(ci)
cj_sinfo = assembler.shape_info(cj)
# Gauss nodes and weights on cell
xi_g, wi_g, phii, dofsi = assembler.shape_eval(ci)
xj_g, wj_g, phij, dofsj = assembler.shape_eval(cj)
#
# Evaluate form
#
form_loc = form.eval((ci,cj), (xi_g,xj_g), \
(wi_g,wj_g), (phii,phij), (dofsi,dofsj))
#
# Define functions
#
u = Nodal(f=lambda x: x, basis=phi)
v = Nodal(f=lambda x: 1 - x, basis=phi)
#
# Get local node values
#
# Degrees of freedom
ci_dofs = phi.dofs(ci)
cj_dofs = phi.dofs(cj)
uj = u.data()[np.array(cj_dofs)]
vi = v.data()[np.array(ci_dofs)]
if ktype == 'symmetric':
# Local form by hand
c10 = 1 / 54
c11 = 1 / 27
c20 = 3 / 2 - 1 - 3 / 2 * 4 / 9 + 8 / 27
c21 = 4 / 27
fl = np.array([[c10 * c20, c10 * c21], [c11 * c20, c11 * c21]])
# Compare computed and explicit local forms
self.assertTrue(np.allclose(fl, form_loc))
# Evaluate Ici Icj k(x,y) y dy (1-x)dx
fa = np.dot(vi.T, form_loc.dot(uj))
fe = vals[ktype]
self.assertAlmostEqual(fa, fe)
def reference_solution():
"""
Use sparse grid method to compute a benchmark solution
"""
mesh = QuadMesh(resolution=(10,10))
mesh.mark_region('boundary', lambda x,y:True,
entity_type='half_edge', on_boundary=True)
element = QuadFE(mesh.dim(), 'Q1')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
n = dofhandler.n_dofs()
phi = Basis(dofhandler, 'u')
phi_x = Basis(dofhandler, 'ux')
phi_y = Basis(dofhandler, 'uy')
yd_fn = Explicit(lambda x: np.sin(2*np.pi*x[:,0])*np.sin(2*np.pi*x[:,1]),dim=2)
g = np.ones((n,1))
#
# Random diffusion coefficient
#
# Sparse grid
tasmanian_library="/home/hans-werner/bin/TASMANIAN-6.0/libtasmaniansparsegrid.so"
grid = TasmanianSG.TasmanianSparseGrid(tasmanian_library=tasmanian_library)
dimensions = 4
outputs = 1
depth = 4
type = 'tensor'
rule = 'gauss-legendre'
grid.makeGlobalGrid(dimensions, outputs, depth, type, rule)
Y = grid.getPoints()
w = grid.getQuadratureWeights()
n_samples = grid.getNumPoints()
x = dofhandler.get_dof_vertices()
a_nodal = 1 + 0.1*(np.outer(np.cos(np.pi*x[:,1]),Y[:,0])+\
np.outer(np.cos(np.pi*x[:,0]),Y[:,1])+\
np.outer(np.sin(2*np.pi*x[:,1]),Y[:,2])+\
np.outer(np.sin(2*np.pi*x[:,0]),Y[:,3]))
a = Nodal(data=a_nodal, dofhandler=dofhandler)
yd_vec = yd_fn.eval(x)
problems = [[Form(a, test=phi_x, trial=phi_x),
Form(a, test=phi_y, trial=phi_y)],
[Form(1, test=phi, trial=phi)]]
assembler = Assembler(problems, mesh)
assembler.assemble()
A = assembler.af[0]['bilinear'].get_matrix()
M = assembler.af[1]['bilinear'].get_matrix()
state = LS(phi)
state.add_dirichlet_constraint('boundary')
adjoint = LS(phi)
adjoint.add_dirichlet_constraint('boundary')
tau = 10
k_max = 20
alpha = 0.1
u = np.zeros((n,1))
norm_dJ_iter = []
J_iter = []
u_iter = []
for k in range(k_max):
print('iteration', k)
#
# Compute average cost and gradient
#
dJ = np.zeros((n,1))
J = 0
print('sampling')
for n in range(n_samples):
print(n)
yn, pn, Jn, dJn = cost_gradient(state,adjoint,A[n],M,
g,u,yd_vec,alpha)
J += w[n]*Jn
dJ += w[n]*dJn
print('')
norm_dJ = np.sqrt(dJ.T.dot(M.dot(dJ)))
#
# Store current iterates
#
norm_dJ_iter.append(norm_dJ)
J_iter.append(J)
u_iter.append(u)
#
# Check for convergence
#
if norm_dJ<1e-8:
break
#
# Update iterate
#
u -= tau*dJ
# -----------------------------------------------------------------------------
# Observations
# -----------------------------------------------------------------------------
# Determine vertices corresponding to production wells
n_prod = 4
h = (x_max-x_min)/(n_prod+2)
x_prod = np.array([(i+1)*h for i in range(n_prod)])
v = dh_y.get_dof_vertices()
dofs_prod = []
for x in x_prod:
dofs_prod.append(np.argmin(abs(v-x)))
#
# Target pressure at production wells
#
z_fn = Explicit(f=lambda x: 3-4*(x[:,0]-1)**2, dim=1, mesh=mesh)
z_data = z_fn.eval(v[dofs_prod])
# -----------------------------------------------------------------------------
# Control
# -----------------------------------------------------------------------------
# Determine the vertices corresponding to the injection wells
n_inj = 6
h = (x_max-x_min)/(n_inj+2)
x_inj = np.array([(i+1)*h for i in range(n_inj)])
dofs_inj = []
for x in x_inj:
dofs_inj.append(np.argmin(abs(v-x)))
u_data = np.zeros((ny,1))
u_data[dofs_inj] = 1
xy = np.array([X.ravel(), Y.ravel()]).T
cells_production = mesh.bin_points(xy)
# Mark vertices
for cell, dummy in cells_production:
cell.get_vertex(2).mark('production')
# Extract degrees of freedom
production_dofs = dh_Q1.get_region_dofs(entity_type='vertex',
entity_flag='production')
v_production = dh_Q1.get_dof_vertices(dofs=production_dofs)
# Target pressure at production wells
z_fn = Explicit(f=lambda x: 3 - 4 * (x[:, 0] - 1)**2 - 8 * (x[:, 1] - 0.5)**2,
dim=2,
mesh=mesh)
y_target = z_fn.eval(v_production)
#
# Locations of injection wells
#
n_injection = (5, 4) # resolution
x_injection = np.linspace(0.5, 1.5, n_injection[0])
y_injection = np.linspace(0.25, 0.75, n_injection[1])
X, Y = np.meshgrid(x_injection, y_injection)
xy = np.array([X.ravel(), Y.ravel()]).T
cells_injection = mesh.bin_points(xy)
# Mark vertices
for cell, dummy in cells_injection:
nt = np.int((t1 - t0) / dt)
# Initial condition
u0 = Constant(0)
# Left Dirichlet condition
def u_left(x):
n_points = x.shape[0]
u = np.zeros((n_points, 1))
left_strip = (x[:, 0] >= 4) * (x[:, 0] <= 6) * (abs(x[:, 1]) < 1e-9)
u[left_strip, 0] = 1
return u
u_left = Explicit(f=u_left, mesh=mesh)
# Define problems
Dx = 1
Dy = 0.1 * Dx
Form(1, trial=v, test=v)
k_max = 1
up = Nodal(f=lambda x: x[:, 1]**2, dofhandler=V1)
um = Nodal(f=lambda x: x[:, 0], dofhandler=V1)
residual = [Form(1, test=v)]
"""
residual = [Form(Kernel(f=up),test=v),
Form(Kernel(f=um), test=v),
Form(Kernel(f=up),test=v_x)]
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()))
def test_constructor(self):
# =====================================================================
# Test 1D
# =====================================================================
#
# Kernel consists of a single explicit Function:
#
f1 = lambda x: x+2
f = Explicit(f1, dim=1)
k = Kernel(f)
x = np.linspace(0,1,100)
n_points = len(x)
# Check that it evaluates correctly.
self.assertTrue(np.allclose(f1(x), k.eval(x).ravel()))
# Check shape of kernel
self.assertEqual(k.eval(x).shape, (n_points,1))
#
# Kernel consists of a combination of two explicit functions
#
f1 = Explicit(lambda x: x+2, dim=1)
f2 = Explicit(lambda x: x**2 + 1, dim=1)
F = lambda f1, f2: f1**2 + f2
f_t = lambda x: (x+2)**2 + x**2 + 1
k = Kernel([f1,f2], F=F)
# Check evaluation
self.assertTrue(np.allclose(f_t(x), k.eval(x).ravel()))
# Check shape
self.assertEqual(k.eval(x).shape, (n_points,1))
#
# Same thing as above, but with nodal functions
#
mesh = Mesh1D(resolution=(1,))
Q1 = QuadFE(1,'Q1')
Q2 = QuadFE(1,'Q2')
dQ1 = DofHandler(mesh,Q1)
dQ2 = DofHandler(mesh,Q2)
# Distribute dofs
[dQ.distribute_dofs() for dQ in [dQ1,dQ2]]
# Basis functions
phi1 = Basis(dQ1,'u')
phi2 = Basis(dQ2,'u')
f1 = Nodal(lambda x: x+2, basis=phi1)
f2 = Nodal(lambda x: x**2 + 1, basis=phi2)
k = Kernel([f1,f2], F=F)
# Check evaluation
self.assertTrue(np.allclose(f_t(x), k.eval(x).ravel()))
#
# Replace f2 above with its derivative
#
k = Kernel([f1,f2], derivatives=['f', 'fx'], F=F)
f_t = lambda x: (x+2)**2 + 2*x
# Check derivative evaluation F = F(f1, df2_dx)
self.assertTrue(np.allclose(f_t(x), k.eval(x).ravel()))
#
# Sampling
#
one = Constant(1)
f1 = Explicit(lambda x: x**2 + 1, dim=1)
# Sampled function
a = np.linspace(0,1,11)
n_samples = len(a)
# Define Dofhandler
dh = DofHandler(mesh, Q2)
dh.distribute_dofs()
dh.set_dof_vertices()
xv = dh.get_dof_vertices()
n_dofs = dh.n_dofs()
phi = Basis(dh, 'u')
# Evaluate parameterized function at mesh dof vertices
f2_m = np.empty((n_dofs, n_samples))
for i in range(n_samples):
f2_m[:,i] = xv.ravel() + a[i]*xv.ravel()**2
f2 = Nodal(data=f2_m, basis=phi)
# Define kernel
F = lambda f1, f2, one: f1 + f2 + one
k = Kernel([f1,f2,one], F=F)
# Evaluate on a fine mesh
x = np.linspace(0,1,100)
n_points = len(x)
self.assertEqual(k.eval(x).shape, (n_points, n_samples))
for i in range(n_samples):
# Check evaluation
self.assertTrue(np.allclose(k.eval(x)[:,i], f1.eval(x)[:,i] + x + a[i]*x**2+ 1))
#
# Sample multiple constant functions
#
f1 = Constant(data=a)
f2 = Explicit(lambda x: 1 + x**2, dim=1)
f3 = Nodal(data=f2_m[:,-1], basis=phi)
F = lambda f1, f2, f3: f1 + f2 + f3
k = Kernel([f1,f2,f3], F=F)
x = np.linspace(0,1,100)
for i in range(n_samples):
self.assertTrue(np.allclose(k.eval(x)[:,i], \
a[i] + f2.eval(x)[:,i] + f3.eval(x)[:,i]))
#
# Submeshes
#
mesh = Mesh1D(resolution=(1,))
mesh_labels = Tree(regular=False)
mesh = Mesh1D(resolution=(1,))
Q1 = QuadFE(1,'Q1')
Q2 = QuadFE(1,'Q2')
dQ1 = DofHandler(mesh,Q1)
dQ2 = DofHandler(mesh,Q2)
# Distribute dofs
[dQ.distribute_dofs() for dQ in [dQ1,dQ2]]
# Basis
p1 = Basis(dQ1)
p2 = Basis(dQ2)
f1 = Nodal(lambda x: x, basis=p1)
f2 = Nodal(lambda x: -2+2*x**2, basis=p2)
one = Constant(np.array([1,2]))
F = lambda f1, f2, one: 2*f1**2 + f2 + one
I = mesh.cells.get_child(0)
kernel = Kernel([f1,f2, one], F=F)
rule1D = GaussRule(5,shape='interval')
x = I.reference_map(rule1D.nodes())
import numpy as np
import matplotlib.pyplot as plt
"""
Error analysis for advection diffusion equation
-epsilon*(d^2u_dx^2 + d^2u_dy^2) + vx*du_dx + vy*du_dy = f
subject to dirichlet boundary conditions.
"""
#
# Define functions
#
# Exact solution
ue = Explicit(f=lambda x: np.sin(x[:, 0]) * np.sin(x[:, 1]),
n_variables=1,
dim=2)
# Forcing function
ffn = lambda x: 2*eps*np.sin(x[:,0])*np.sin(x[:,1]) + \
x[:,0]*np.cos(x[:,0])*np.sin(x[:,1]) + \
x[:,1]*np.sin(x[:,0])*np.cos(x[:,1])
f = Explicit(ffn, dim=2)
# Velocity function
vx = Explicit(lambda x: x[:, 0], dim=2)
vy = Explicit(lambda x: x[:, 1], dim=2)
errors = {}
for resolution in [(5, 5), (10, 10), (20, 20), (40, 40)]:
#
Q1 = QuadFE(2,'Q1') # element for pressure
Q2 = QuadFE(2,'Q2') # element for velocity
# Dofhandler
DQ1 = DofHandler(mesh, Q1)
DQ2 = DofHandler(mesh, Q2)
# Time discretization
T = 1
nt = 101
t = np.linspace(0,T,nt)
# Problem Parameters
g = 10
gma = 0.1
U = 0.1
tht = 1
nu = Explicit(f=lambda x: np.sin(x[:,0])**2+1, dim=2)
# Explicit solution
u1 = Explicit(f=lambda x,t: t[:,0]**3*np.sin(np.pi*x[:,0])*np.sin(np.pi*x[:,1]),\
dim=2, n_variables=2)
#
# Iterate over time ss
#
u = np.empty()
for i in range(nt):
pass
