def test_system_2d():
domain = Square()
V = FunctionSpace('V', domain)
x, y = V.coordinates
u, v, p, q = [TestFunction(V, name=i) for i in ['u', 'v', 'p', 'q']]
a1, a2, b1, b2 = [Constant(i, real=True) for i in ['a1', 'a2', 'b1', 'b2']]
# ...
a = BilinearForm((v, u), dot(grad(u), grad(v)))
m = BilinearForm((v, u), u * v)
expr = a(p, u) + a1 * m(p, u) + b1 * m(p, v) + a(
q, v) + a2 * m(q, u) + b2 * m(q, v)
b = BilinearForm(((p, q), (u, v)), expr)
print(evaluate(b, verbose=True))
# ...
# ...
f1 = x * y
f2 = x + y
l1 = LinearForm(p, f1 * p)
l2 = LinearForm(q, f2 * q)
expr = l1(p) + l2(q)
l = LinearForm((p, q), expr)
print(evaluate(l, verbose=True))
def test_boundary_2d_2():
Omega_1 = InteriorDomain('Omega_1', dim=2)
B1 = Boundary('B1', Omega_1)
B2 = Boundary('B2', Omega_1)
B3 = Boundary('B3', Omega_1)
domain = Domain('Omega', interiors=[Omega_1], boundaries=[B1, B2, B3])
V = FunctionSpace('V', domain)
v = TestFunction(V, name='v')
u = TestFunction(V, name='u')
x, y = V.coordinates
alpha = Constant('alpha')
# ...
print('==== l0 ====')
l0 = LinearForm(v, x * y * v, name='l0')
print(evaluate(l0, verbose=VERBOSE))
print('')
# ...
# ...
print('==== l1 ====')
g = Tuple(x**2, y**2)
l1 = LinearForm(v, v * trace_1(g, domain.boundary))
print(evaluate(l1, verbose=VERBOSE))
print('')
# ...
# ...
print('==== l2 ====')
B_neumann = Union(B1, B2)
g = Tuple(x**2, y**2)
l2 = LinearForm(v, v * trace_1(g, B_neumann), name='l2')
print(evaluate(l2, verbose=VERBOSE))
print('')
# ...
# ...
print('==== l3 ====')
l3 = LinearForm(v, l2(v))
assert (l3(v).__str__ == l2(v).__str__)
print(evaluate(l3, verbose=VERBOSE))
print('')
# ...
# ...
print('==== l4 ====')
l4 = LinearForm(v, l0(v) + l2(v))
print(evaluate(l4, verbose=VERBOSE))
print('')
def test_user_function_2d_1():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
# right hand side
f = Function('f')
V = FunctionSpace('V', domain)
u, v = [TestFunction(V, name=i) for i in ['u', 'v']]
# ...
expr = dot(grad(u), grad(v)) + f(x, y) * u * v
a = BilinearForm((v, u), expr)
print(a)
print(evaluate(a, verbose=True))
print('')
# ...
# ...
expr = f(x, y) * v
l = LinearForm(v, expr)
print(l)
print(evaluate(l, verbose=True))
print('')
def test_evaluation_2d_1():
domain = Domain('Omega', dim=2)
B_neumann = Boundary(r'\Gamma_1', domain)
V = FunctionSpace('V', domain)
W = VectorFunctionSpace('W', domain)
p, q = [TestFunction(V, name=i) for i in ['p', 'q']]
u, v = [VectorTestFunction(W, name=i) for i in ['u', 'v']]
alpha = Constant('alpha')
x, y = V.coordinates
F = Field('F', space=V)
a1 = BilinearForm((p, q), dot(grad(p), grad(q)))
m = BilinearForm((p, q), p * q)
a2 = BilinearForm((p, q), a1(p, q) + alpha * m(p, q))
a3 = BilinearForm((u, v), rot(u) * rot(v) + alpha * div(u) * div(v))
a11 = BilinearForm((v, u), inner(grad(v), grad(u)))
a12 = BilinearForm((v, p), div(v) * p)
a4 = BilinearForm(((v, q), (u, p)), a11(v, u) - a12(v, p) + a12(u, q))
l0 = LinearForm(p, F * p)
l_neu = LinearForm(p, p * trace_1(grad(F), B_neumann))
l = LinearForm(p, l0(p) + l_neu(p))
# ...
print(a1)
print(evaluate(a1))
print('')
# ...
# ...
print(a2)
print(evaluate(a2))
print('')
# ...
# ...
print(a3)
print(evaluate(a3))
print('')
# ...
# ...
print(a4)
print(evaluate(a4))
print('')
# ...
# ...
print(l)
print(evaluate(l))
print('')
def test_calls_2d_3():
domain = Square()
V = FunctionSpace('V', domain)
x, y = domain.coordinates
pn = Field('pn', V)
wn = Field('wn', V)
dp = TestFunction(V, name='dp')
dw = TestFunction(V, name='dw')
tau = TestFunction(V, name='tau')
sigma = TestFunction(V, name='sigma')
Re = Constant('Re', real=True)
dt = Constant('dt', real=True)
alpha = Constant('alpha', real=True)
l1 = LinearForm(tau,
bracket(pn, wn) * tau - 1. / Re * dot(grad(tau), grad(wn)))
# ...
l = LinearForm((tau, sigma), dt * l1(tau))
print(evaluate(l, verbose=True))
def test_vector_2d_1():
domain = Domain('Omega', dim=DIM)
W1 = VectorFunctionSpace('W1', domain)
T1 = VectorFunctionSpace('T1', domain)
w1 = VectorTestFunction(W1, name='w1')
t1 = VectorTestFunction(T1, name='t1')
x, y = W1.coordinates
F = VectorField(W1, 'F')
# # ...
# l1 = LinearForm(w1, dot(w1, F), name='l1')
# print(l1)
# print(atomize(l1))
# print(evaluate(l1))
# print('')
# # ...
#
# # ...
# l2 = LinearForm(w1, rot(w1)*rot(F) + div(w1)*div(F), name='l2')
# print(l2)
# print(atomize(l2))
# print(evaluate(l2))
# print('')
# # ...
# ...
f = Tuple(sin(pi * x) * sin(pi * y), sin(pi * x) * sin(pi * y))
error = Matrix([F[0] - f[0], F[1] - f[1]])
l2_norm = Norm(error, domain, kind='l2')
print(l2_norm)
print(atomize(l2_norm))
print(evaluate(l2_norm))
print('')
# ...
# ...
f = Tuple(sin(pi * x) * sin(pi * y), sin(pi * x) * sin(pi * y))
error = Matrix([F[0] - f[0], F[1] - f[1]])
h1_norm = Norm(error, domain, kind='h1')
print(h1_norm)
print(atomize(h1_norm))
print(evaluate(h1_norm))
print('')
def test_calls_2d_2():
domain = Square()
V = FunctionSpace('V', domain)
x, y = V.coordinates
u, v = [TestFunction(V, name=i) for i in ['u', 'v']]
Un = Field('Un', V)
# ...
a = BilinearForm((v, u), dot(grad(u), grad(v)))
expr = a(v, Un)
print(evaluate(expr, verbose=True))
# ...
# ...
l = LinearForm(v, a(v, Un))
print(evaluate(l, verbose=True))
def test_evaluation_2d_2():
domain = Square()
x, y = domain.coordinates
f0 = Tuple(2 * pi**2 * sin(pi * x) * sin(pi * y),
2 * pi**2 * sin(pi * x) * sin(pi * y))
f1 = cos(pi * x) * cos(pi * y)
W = VectorFunctionSpace('W', domain)
V = FunctionSpace('V', domain)
X = ProductSpace(W, V)
# TODO improve: naming are not given the same way
F = VectorField(W, name='F')
G = Field('G', V)
u, v = [VectorTestFunction(W, name=i) for i in ['u', 'v']]
p, q = [TestFunction(V, name=i) for i in ['p', 'q']]
a0 = BilinearForm((v, u), inner(grad(v), grad(u)))
a1 = BilinearForm((q, p), p * q)
a = BilinearForm(((v, q), (u, p)), a0(v, u) + a1(q, p))
l0 = LinearForm(v, dot(f0, v))
l1 = LinearForm(q, f1 * q)
l = LinearForm((v, q), l0(v) + l1(q))
# ...
print(a)
print(evaluate(a))
print('')
# ...
# ...
print(l)
print(evaluate(l))
print('')
def test_curldiv_2d():
domain = Square()
W1 = VectorFunctionSpace('W1', domain)
T1 = VectorFunctionSpace('T1', domain)
w1 = VectorTestFunction(W1, name='w1')
t1 = VectorTestFunction(T1, name='t1')
mu = Constant('mu')
# ...
a = BilinearForm((w1, t1),
rot(w1) * rot(t1) + mu * div(w1) * div(t1),
name='a')
print(a)
print(atomize(a))
print(evaluate(a))
def test_boundary_2d_1():
domain = Domain('Omega', dim=DIM)
V1 = FunctionSpace('V1', domain)
V2 = FunctionSpace('V2', domain)
U1 = FunctionSpace('U1', domain)
U2 = FunctionSpace('U2', domain)
W1 = VectorFunctionSpace('W1', domain)
W2 = VectorFunctionSpace('W2', domain)
T1 = VectorFunctionSpace('T1', domain)
T2 = VectorFunctionSpace('T2', domain)
v1 = TestFunction(V1, name='v1')
v2 = TestFunction(V2, name='v2')
u1 = TestFunction(U1, name='u1')
u2 = TestFunction(U2, name='u2')
w1 = VectorTestFunction(W1, name='w1')
w2 = VectorTestFunction(W2, name='w2')
t1 = VectorTestFunction(T1, name='t1')
t2 = VectorTestFunction(T2, name='t2')
x, y = V1.coordinates
alpha = Constant('alpha')
B1 = Boundary(r'\Gamma_1', domain)
B2 = Boundary(r'\Gamma_2', domain)
B3 = Boundary(r'\Gamma_3', domain)
# ...
with pytest.raises(UnconsistentError):
expr = dot(grad(v1), grad(u1)) + v1 * trace_0(u1, B1)
a = BilinearForm((v1, u1), expr, name='a')
# ...
# ...
with pytest.raises(UnconsistentError):
expr = v1 * trace_0(u1, B3) + v1 * trace_1(grad(u1),
B3) + u1 * trace_0(v1, B2)
a1 = BilinearForm((v1, u1), expr, name='a1')
# ...
# ...
expr = dot(grad(v1), grad(u1))
a_0 = BilinearForm((v1, u1), expr, name='a_0')
expr = v1 * trace_0(u1, B1)
a_bnd = BilinearForm((v1, u1), expr, name='a_bnd')
expr = a_0(v1, u1) + a_bnd(v1, u1)
a = BilinearForm((v1, u1), expr, name='a')
print(a)
print(evaluate(a, verbose=True))
print('')
# import sys; sys.exit(0)
# ...
# ...
expr = v1 * trace_0(u1, B1) + v1 * trace_1(grad(u1), B1)
a1 = BilinearForm((v1, u1), expr, name='a1')
expr = u1 * trace_1(grad(v1), B2)
a2 = BilinearForm((v1, u1), expr, name='a2')
expr = a1(v2, u2) + a2(v2, u2)
# as expected, we can define the form call, but we cannot create a Bilinear
# form out of it.
# TODO add assert on exception type
# a = BilinearForm((v2, u2), expr, name='a')
print(expr)
print('')
# ...
# ...
expr = v1 * trace_0(u1, B1)
a0 = BilinearForm((v1, u1), expr, name='a0')
expr = v1 * trace_1(grad(u1), B1)
a1 = BilinearForm((v1, u1), expr, name='a1')
expr = v1 * trace_1(grad(u1), B2)
a2 = BilinearForm((v1, u1), expr, name='a2')
expr = dot(grad(u1), grad(v1))
a3 = BilinearForm((v1, u1), expr, name='a3')
expr = u1 * v1
a4 = BilinearForm((v1, u1), expr, name='a4')
# TODO Mul not treated yet
# expr = a0(v2, u2) + a1(v2, u2) + alpha * a2(v2, u2) + a3(v2, u2) + alpha*a4(v2, u2)
# a = BilinearForm((v2, u2), expr, name='a')
## print(expr)
# print(evaluate(expr, verbose=True))
# print('')
#
# print(evaluate(a, verbose=True))
# print('')
# expr = a(v2, u2) + a1(v2, u2)
# b = BilinearForm((v2, u2), expr, name='b')
# print(b)
# print(evaluate(b, verbose=True))
# ...
# ...
g = Tuple(x**2, y**2)
expr = v1 * trace_1(g, B1)
l1 = LinearForm(v1, expr, name='l1')
print(l1)
# print(atomize(l1))
# print(evaluate(l1))
print('')
def test_norm_2d():
domain = Domain('Omega', dim=DIM)
x, y = domain.coordinates
V = FunctionSpace('V', domain)
F = Field('F', V)
# ...
expr = x * y
l2_norm_u = Norm(expr, domain, kind='l2')
h1_norm_u = Norm(expr, domain, kind='h1')
print('> l2 norm = ', evaluate(l2_norm_u))
print('> h1 norm = ', evaluate(h1_norm_u))
print('')
# ...
# ...
expr = sin(pi * x) * sin(pi * y)
l2_norm_u = Norm(expr, domain, kind='l2')
h1_norm_u = Norm(expr, domain, kind='h1')
print('> l2 norm = ', evaluate(l2_norm_u))
print('> h1 norm = ', evaluate(h1_norm_u))
print('')
# ...
# ...
expr = F - x * y
l2_norm_u = Norm(expr, domain, kind='l2')
h1_norm_u = Norm(expr, domain, kind='h1')
print('> l2 norm = ', evaluate(l2_norm_u))
print('> h1 norm = ', evaluate(h1_norm_u))
print('')
# ...
# ...
expr = F - sin(pi * x) * sin(pi * y)
l2_norm_u = Norm(expr, domain, kind='l2')
h1_norm_u = Norm(expr, domain, kind='h1')
print('> l2 norm = ', evaluate(l2_norm_u))
print('> h1 norm = ', evaluate(h1_norm_u))
print('')
# ...
# ...
expr = F - sin(0.5 * pi * (1. - x)) * sin(pi * y)
l2_norm_u = Norm(expr, domain, kind='l2')
h1_norm_u = Norm(expr, domain, kind='h1')
print('> l2 norm = ', evaluate(l2_norm_u))
print('> h1 norm = ', evaluate(h1_norm_u))
print('')
# ...
# ...
expr = F - cos(0.5 * pi * x) * sin(pi * y)
l2_norm_u = Norm(expr, domain, kind='l2')
h1_norm_u = Norm(expr, domain, kind='h1')
print('> l2 norm = ', evaluate(l2_norm_u))
print('> h1 norm = ', evaluate(h1_norm_u))
print('')
def test_calls_2d():
domain = Domain('Omega', dim=DIM)
V1 = FunctionSpace('V1', domain)
V2 = FunctionSpace('V2', domain)
U1 = FunctionSpace('U1', domain)
U2 = FunctionSpace('U2', domain)
W1 = VectorFunctionSpace('W1', domain)
W2 = VectorFunctionSpace('W2', domain)
T1 = VectorFunctionSpace('T1', domain)
T2 = VectorFunctionSpace('T2', domain)
v1 = TestFunction(V1, name='v1')
v2 = TestFunction(V2, name='v2')
u1 = TestFunction(U1, name='u1')
u2 = TestFunction(U2, name='u2')
w1 = VectorTestFunction(W1, name='w1')
w2 = VectorTestFunction(W2, name='w2')
t1 = VectorTestFunction(T1, name='t1')
t2 = VectorTestFunction(T2, name='t2')
V = ProductSpace(V1, V2)
U = ProductSpace(U1, U2)
x, y = V1.coordinates
alpha = Constant('alpha')
F = Field('F', space=V1)
# ...
a1 = BilinearForm((v1, u1), u1 * v1, name='a1')
print(a1)
print(atomize(a1))
print(evaluate(a1))
print('')
expr = a1(v2, u2)
a = BilinearForm((v2, u2), expr, name='a')
print(a)
print(atomize(a))
print(evaluate(a))
print('')
# ...
# ...
a = BilinearForm((v1, u1), dot(grad(v1), grad(u1)), name='a')
print(a)
print(atomize(a))
print(evaluate(a))
print('')
# ...
# ...
a1 = BilinearForm((v1, u1), dot(grad(v1), grad(u1)), name='a1')
expr = a1(v2, u2)
a = BilinearForm((v2, u2), expr, name='a')
print(a)
print(atomize(a))
print(evaluate(a))
print('')
# ...
# ...
a1 = BilinearForm((v1, u1), u1 * v1, name='a1')
a2 = BilinearForm((v1, u1), dx(u1) * dx(v1), name='a2')
expr = a1(v2, u2) + a2(v2, u2)
a = BilinearForm((v2, u2), expr, name='a')
print(a)
print(atomize(a))
print(evaluate(a))
print('')
# ...
# ...
a1 = BilinearForm((v1, u1), u1 * v1, name='a1')
a2 = BilinearForm((v1, u1), dx(u1) * dx(v1), name='a2')
expr = a1(v1, u2)
print('> before = ', expr)
expr = expr.subs(u2, u1)
print('> after = ', expr)
print('')
expr = a1(v1, u2) + a1(v2, u2)
print('> before = ', expr)
expr = expr.subs(u2, u1)
print('> after = ', expr)
print('')
# ...
# ...
a1 = BilinearForm((v1, u1), u1 * v1, name='a1')
a2 = BilinearForm((v1, u1), dx(u1) * dx(v1), name='a2')
expr = a1(v1, u2) + a2(v2, u1)
a = BilinearForm(((v1, v2), (u1, u2)), expr, name='a')
print(a)
print(atomize(a))
print(evaluate(a))
print('')
# ...
# ...
a = BilinearForm((w1, t1), rot(w1) * rot(t1) + div(w1) * div(t1), name='a')
print(a)
print(atomize(a))
print(evaluate(a))
# ...
# ...
a1 = BilinearForm((v1, u1), u1 * v1, name='a1')
a2 = BilinearForm((v1, u1), dx(u1) * dx(v1), name='a2')
a3 = BilinearForm((w1, t1),
rot(w1) * rot(t1) + div(w1) * div(t1),
name='a3')
a4 = BilinearForm((w1, u1), div(w1) * u1, name='a4')
expr = a3(w2, t2) + a2(v2, u2) + a4(w2, u2)
a = BilinearForm(((w2, v2), (t2, u2)), expr, name='a')
print(a)
print(atomize(a))
print(evaluate(a))
# ...
# ...
a1 = BilinearForm((v1, u1), laplace(u1) * laplace(v1), name='a1')
print(a1)
print(atomize(a1))
print(evaluate(a1))
print('')
# ...
# ...
a1 = BilinearForm((v1, u1), inner(hessian(u1), hessian(v1)), name='a1')
print('================================')
print(a1)
print(atomize(a1))
print(evaluate(a1))
print('')
# ...
# ...
l1 = LinearForm(v1, x * y * v1, name='11')
expr = l1(v2)
l = LinearForm(v2, expr, name='1')
print(l)
print(atomize(l))
print(evaluate(l))
# ...
# ...
l1 = LinearForm(v1, x * y * v1, name='l1')
l2 = LinearForm(v2, cos(x + y) * v2, name='l2')
expr = l1(u1) + l2(u2)
l = LinearForm((u1, u2), expr, name='1')
print(l)
print(atomize(l))
print(evaluate(l))
# ...
# ...
l1 = LinearForm(v1, x * y * v1, name='l1')
l2 = LinearForm(v2, cos(x + y) * v2, name='l2')
expr = l1(u1) + alpha * l2(u2)
l = LinearForm((u1, u2), expr, name='1')
print(l)
print(atomize(l))
print(evaluate(l))
# ...
# ...
l1 = LinearForm(v1, x * y * v1, name='l1')
l2 = LinearForm(w1, div(w1), name='l2')
expr = l1(v2) + l2(w2)
l = LinearForm((v2, w2), expr, name='1')
print(l)
print(atomize(l))
print(evaluate(l))
# ...
# ...
I1 = Integral(x * y, domain, name='I1')
print(I1)
print(atomize(I1))
print(evaluate(I1))
# ...
# ...
expr = F - cos(2 * pi * x) * cos(3 * pi * y)
expr = dot(grad(expr), grad(expr))
I2 = Integral(expr, domain, name='I2')
print(I2)
print(atomize(I2))
print(evaluate(I2))
# ...
# ...
expr = F - cos(2 * pi * x) * cos(3 * pi * y)
expr = dot(grad(expr), grad(expr))
I2 = Integral(expr, domain, name='I2')
print(I2)
print(atomize(I2))
print(evaluate(I2))
# ...
# ... stokes
V = VectorFunctionSpace('V', domain)
W = FunctionSpace('W', domain)
v = VectorTestFunction(V, name='v')
u = VectorTestFunction(V, name='u')
p = TestFunction(W, name='p')
q = TestFunction(W, name='q')
a = BilinearForm((v, u), inner(grad(v), grad(u)), name='a')
b = BilinearForm((v, p), div(v) * p, name='b')
A = BilinearForm(((v, q), (u, p)), a(v, u) - b(v, p) + b(u, q), name='A')
print(A)
print(atomize(A))
print(evaluate(A))
def test_stabilization_2d_1():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
b1 = 1.
b2 = 0.
b = Tuple(b1, b2)
# right hand side
f = x * y
e = ElementDomain()
area = Area(e)
V = FunctionSpace('V', domain)
u, v = [TestFunction(V, name=i) for i in ['u', 'v']]
# ...
expr = kappa * dot(grad(u), grad(v)) + dot(b, grad(u)) * v
a = BilinearForm((v, u), expr)
# ...
# ...
expr = f * v
l = LinearForm(v, expr)
# ...
# ...
expr = (-kappa * laplace(u) + dot(b, grad(u))) * dot(b, grad(v))
s1 = BilinearForm((v, u), expr)
expr = -f * dot(b, grad(v))
l1 = LinearForm(v, expr)
# ...
# ...
expr = (-kappa * laplace(u) + dot(b, grad(u))) * (dot(b, grad(v)) -
kappa * laplace(v))
s2 = BilinearForm((v, u), expr)
expr = -f * (dot(b, grad(v)) - kappa * laplace(v))
l2 = LinearForm(v, expr)
# ...
# ...
expr = (-kappa * laplace(u) + dot(b, grad(u))) * (dot(b, grad(v)) +
kappa * laplace(v))
s3 = BilinearForm((v, u), expr)
expr = -f * (dot(b, grad(v)) + kappa * laplace(v))
l3 = LinearForm(v, expr)
# ...
# ...
expr = a(v, u) + mu * area * s1(v, u)
a1 = BilinearForm((v, u), expr)
# ...
# ...
expr = a(v, u) + mu * area * s2(v, u)
a2 = BilinearForm((v, u), expr)
# ...
# ...
expr = a(v, u) + mu * area * s3(v, u)
a3 = BilinearForm((v, u), expr)
# ...
print(a1)
print(evaluate(a1, verbose=True))
print('')
print(a2)
print(evaluate(a2, verbose=True))
print('')
print(a3)
print(evaluate(a3, verbose=True))
print('')