def test_linearize_form_2d_3():
"""steady Euler equation."""
domain = Domain('Omega', dim=2)
U = VectorFunctionSpace('U', domain)
W = ScalarFunctionSpace('W', domain)
# Test functions
v = element_of(U, name='v')
phi = element_of(W, name='phi')
q = element_of(W, name='q')
# Steady-state fields
U_0 = element_of(U, name='U_0')
Rho_0 = element_of(W, name='Rho_0')
P_0 = element_of(W, name='P_0')
# Trial functions (displacements from steady-state)
d_u = element_of(U, name='d_u')
d_rho = element_of(W, name='d_rho')
d_p = element_of(W, name='d_p')
# Shortcut
int_0 = lambda expr: integral(domain, expr)
# The Euler equations are a system of three non-linear equations; for each of
# them we create a linear form in the test functions (phi, v, q) respectively.
e1 = div(Rho_0 * U_0)
l1 = LinearForm(phi, int_0(e1 * phi))
e2 = Rho_0 * convect(U_0, U_0) + grad(P_0)
l2 = LinearForm(v, int_0(dot(e2, v)))
e3 = div(P_0 * U_0)
l3 = LinearForm(q, int_0(e3 * q))
# ...
# Linearize l1, l2 and l3 separately
a1 = linearize(l1, fields=[Rho_0, U_0], trials=[d_rho, d_u])
a2 = linearize(l2, fields=[Rho_0, U_0, P_0], trials=[d_rho, d_u, d_p])
a3 = linearize(l3, fields=[U_0, P_0], trials=[d_u, d_p])
# Check individual bilinear forms
d_e1 = div(U_0 * d_rho + Rho_0 * d_u)
d_e2 = d_rho * convect(U_0, U_0) + \
Rho_0 * convect(d_u, U_0) + \
Rho_0 * convect(U_0, d_u) + grad(d_p)
d_e3 = div(d_p * U_0 + P_0 * d_u)
assert a1([d_rho, d_u], phi) == int_0(d_e1 * phi)
assert a2([d_rho, d_u, d_p], v) == int_0(dot(d_e2, v))
assert a3([d_u, d_p], q) == int_0(d_e3 * q)
# Linearize linear form of system: l = l1 + l2 + l3
l = LinearForm((phi, v, q), l1(phi) + l2(v) + l3(q))
a = linearize(l, fields=[Rho_0, U_0, P_0], trials=[d_rho, d_u, d_p])
# Check composite linear form
assert a([d_rho, d_u, d_p], [phi, v, q]) == \
int_0(d_e1 * phi + dot(d_e2, v) + d_e3 * q)
def test_linearize_form_2d_4():
domain = Domain('Omega', dim=2)
Gamma_N = Boundary(r'\Gamma_N', domain)
x, y = domain.coordinates
V = ScalarFunctionSpace('V', domain)
v = element_of(V, name='v')
u = element_of(V, name='u')
du = element_of(V, name='du')
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(Gamma_N, expr)
# g = Matrix((cos(pi*x)*sin(pi*y),
# sin(pi*x)*cos(pi*y)))
expr = dot(grad(v), grad(u)) - 4. * exp(-u) * v # + v*trace_1(g, Gamma_N)
l = LinearForm(v, int_0(expr))
# linearising l around u, using du
a = linearize(l, u, trials=du)
assert a(du, v) == int_0(dot(grad(v), grad(du)) + 4. * exp(-u) * du * v)
def test_linearize_form_2d_2():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
v, F, u = elements_of(V, names='v, F, u')
int_0 = lambda expr: integral(domain, expr)
# ...
l1 = LinearForm(v, int_0(F**2 * v))
l = LinearForm(v, l1(v))
a = linearize(l, F, trials=u)
expected = linearize(l1, F, trials=u)
assert a == expected
def test_linearize_form_2d_3():
"""steady Euler equation."""
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
U = VectorFunctionSpace('U', domain)
W = FunctionSpace('W', domain)
v = VectorTestFunction(U, name='v')
phi = TestFunction(W, name='phi')
q = TestFunction(W, name='q')
U_0 = VectorField(U, name='U_0')
Rho_0 = Field(W, name='Rho_0')
P_0 = Field(W, name='P_0')
# ...
expr = div(Rho_0 * U_0) * phi
l1 = LinearForm(phi, expr)
expr = Rho_0 * dot(convect(U_0, grad(U_0)), v) + dot(grad(P_0), v)
l2 = LinearForm(v, expr)
expr = dot(U_0, grad(P_0)) * q + P_0 * div(U_0) * q
l3 = LinearForm(q, expr)
# ...
a1 = linearize(l1, [Rho_0, U_0], trials=['d_rho', 'd_u'])
print(a1)
print('')
a2 = linearize(l2, [Rho_0, U_0, P_0], trials=['d_rho', 'd_u', 'd_p'])
print(a2)
print('')
a3 = linearize(l3, [P_0, U_0], trials=['d_p', 'd_u'])
print(a3)
print('')
l = LinearForm((phi, v, q), l1(phi) + l2(v) + l3(q))
a = linearize(l, [Rho_0, U_0, P_0], trials=['d_rho', 'd_u', 'd_p'])
print(a)
export(a, 'steady_euler.png')
def test_linearize_expr_2d_1():
domain = Domain('Omega', dim=2)
x,y = domain.coordinates
V1 = ScalarFunctionSpace('V1', domain)
W1 = VectorFunctionSpace('W1', domain)
v1 = element_of(V1, name='v1')
w1 = element_of(W1, name='w1')
alpha = Constant('alpha')
F = element_of(V1, name='F')
G = element_of(W1, 'G')
# ...
l = LinearExpr(v1, F**2*v1)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(v1, dot(grad(F), grad(F))*v1)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(v1, exp(-F)*v1)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(v1, cos(F)*v1)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(v1, cos(F**2)*v1)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(v1, F**2*dot(grad(F), grad(v1)))
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(w1, dot(rot(G), grad(G))*w1)
a = linearize(l, G, trials='u1')
print(a)
def test_linearize_expr_2d_2():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
V1 = ScalarFunctionSpace('V1', domain)
v1 = element_of(V1, name='v1')
alpha = Constant('alpha')
F = element_of(V1, name='F')
G = element_of(V1, name='G')
# ...
l1 = LinearExpr(v1, F**2 * v1)
l = LinearExpr(v1, l1(v1))
a = linearize(l, F, trials='u1')
print(a)
expected = linearize(l1, F, trials='u1')
assert (linearize(l, F, trials='u1') == expected)
def test_linearize_form_2d_1():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
W = VectorFunctionSpace('W', domain)
v, F, u = elements_of(V, names='v, F, u')
w, G, m = elements_of(W, names='w, G, m')
int_0 = lambda expr: integral(domain, expr)
# ...
l = LinearForm(v, int_0(F**2 * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(2 * F * u * v)
# ...
# ...
l = LinearForm(v, int_0(dot(grad(F), grad(F)) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(2 * dot(grad(F), grad(u)) * v)
# ...
# ...
l = LinearForm(v, int_0(exp(-F) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(-exp(-F) * u * v)
# ...
# ...
l = LinearForm(v, int_0(cos(F) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(-sin(F) * u * v)
# ...
# ...
l = LinearForm(v, int_0(cos(F**2) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(-2 * F * sin(F**2) * u * v)
# ...
# ...
l = LinearForm(v, int_0(F**2 * dot(grad(F), grad(v))))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(2 * F * u * dot(grad(F), grad(v)) +
F**2 * dot(grad(u), grad(v)))
# ...
# ...
l = LinearForm(w, int_0(dot(rot(G), grad(G)) * w))
a = linearize(l, G, trials=m)
assert a(m, w) == int_0((dot(rot(m), grad(G)) + dot(rot(G), grad(m))) * w)