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_latex_2d_5():
DIM = 2
domain = Domain('Omega', dim=DIM)
# ... abstract model
W1 = VectorFunctionSpace('W1', domain)
w1 = element_of(W1, name='w1')
F = element_of(W1, 'F')
int_0 = lambda expr: integral(domain, expr)
# ...
l1 = LinearForm(w1, int_0(dot(w1, F)))
print(latex(l1))
print('')
# ...
# ...
l2 = LinearForm(w1, int_0(rot(w1) * rot(F) + div(w1) * div(F)))
print(latex(l2))
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_calculus_2d_1():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
W = VectorFunctionSpace('W', domain)
alpha, beta, gamma = [Constant(i) for i in ['alpha', 'beta', 'gamma']]
f, g, h = elements_of(V, names='f, g, h')
F, G, H = elements_of(W, names='F, G, H')
# ... scalar gradient properties
assert (grad(f + g) == grad(f) + grad(g))
assert (grad(alpha * h) == alpha * grad(h))
assert (grad(alpha * f + beta * g) == alpha * grad(f) + beta * grad(g))
assert (grad(f * g) == f * grad(g) + g * grad(f))
assert (grad(f / g) == -f * grad(g) / g**2 + grad(f) / g)
assert (expand(grad(f * g * h)) == f * g * grad(h) + f * h * grad(g) +
g * h * grad(f))
# ...
# ... vector gradient properties
assert (grad(F + G) == grad(F) + grad(G))
assert (grad(alpha * H) == alpha * grad(H))
assert (grad(alpha * F + beta * G) == alpha * grad(F) + beta * grad(G))
assert (grad(dot(F, G)) == convect(F, G) + convect(G, F) +
cross(F, curl(G)) - cross(curl(F), G))
# ...
# ... curl properties
assert (curl(f + g) == curl(f) + curl(g))
assert (curl(alpha * h) == alpha * curl(h))
assert (curl(alpha * f + beta * g) == alpha * curl(f) + beta * curl(g))
# ...
# ... laplace properties
assert (laplace(f + g) == laplace(f) + laplace(g))
assert (laplace(alpha * h) == alpha * laplace(h))
assert (laplace(alpha * f + beta * g) == alpha * laplace(f) +
beta * laplace(g))
# ...
# ... divergence properties
assert (div(F + G) == div(F) + div(G))
assert (div(alpha * H) == alpha * div(H))
assert (div(alpha * F + beta * G) == alpha * div(F) + beta * div(G))
assert (div(cross(F, G)) == -dot(F, curl(G)) + dot(G, curl(F)))
# ...
# ... rot properties
assert (rot(F + G) == rot(F) + rot(G))
assert (rot(alpha * H) == alpha * rot(H))
assert (rot(alpha * F + beta * G) == alpha * rot(F) + beta * rot(G))
def test_tensorize_2d_2():
domain = Domain('Omega', dim=2)
V = VectorFunctionSpace('V', domain)
u, v = elements_of(V, names='u, v')
int_0 = lambda expr: integral(domain , expr)
# ...
# a = BilinearForm((u,v), dot(u,v))
a = BilinearForm((u,v), int_0(curl(u)*curl(v) + div(u)*div(v)))
expr = TensorExpr(a, domain=domain)
print(expr)
def test_projector_2d_1():
DIM = 2
domain = Domain('Omega', dim=DIM)
V = ScalarFunctionSpace('V', domain, kind=None)
W = VectorFunctionSpace('W', domain, kind=None)
v, w = element_of(V * W, ['v', 'w'])
# ...
P_V = Projector(V)
assert (P_V.space == V)
Pv = P_V(v)
assert (isinstance(Pv, ScalarTestFunction))
assert (Pv == v)
assert (grad(Pv**2) == 2 * v * grad(v))
Pdiv_w = P_V(div(w))
assert (isinstance(Pdiv_w, ScalarTestFunction))
# ...
# ...
P_W = Projector(W)
assert (P_W.space == W)
Pw = P_W(w)
assert (isinstance(Pw, VectorTestFunction))
assert (Pw == w)
Pgrad_v = P_W(grad(v))
assert (isinstance(Pgrad_v, VectorTestFunction))
assert (P_W(Pgrad_v) == Pgrad_v)
def test_latex_2d_2():
DIM = 2
domain = Domain('Omega', dim=DIM)
V = VectorFunctionSpace('V', domain)
x, y = V.coordinates
v = element_of(V, name='v')
u = element_of(V, name='u')
# F = element_of(V, name='F')
int_0 = lambda expr: integral(domain, expr)
assert (latex(v) == r'\mathbf{v}')
assert (latex(inner(
grad(v), grad(u))) == r'\nabla{\mathbf{u}} : \nabla{\mathbf{v}}')
a = BilinearForm((v, u), int_0(inner(grad(v), grad(u))))
print(latex(a))
# assert(latex(a) == r'\int_{0}^{1}\int_{0}^{1} \nabla{\mathbf{v}} : \nabla{\mathbf{u}} dxdy')
b = LinearForm(v, int_0(sin(pi * x) * cos(pi * y) * div(v)))
print(latex(b))
def test_terminal_expr_linear_2d_2():
domain = Domain('Omega', dim=2)
B1 = Boundary(r'\Gamma_1', domain)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
V = VectorFunctionSpace('V', domain)
u, u1, u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
# ...
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
g = Matrix((x, y))
l = LinearForm(v, int_0(dot(g, v)))
print(TerminalExpr(l))
print('')
# ...
# ...
g = Matrix((x, y))
l = LinearForm(v, int_0(dot(g, v) + div(v)))
print(TerminalExpr(l))
print('')
def test_tensorize_2d_2_mapping():
DIM = 2
M = Mapping('M', DIM)
domain = Domain('Omega', dim=DIM)
V = VectorFunctionSpace('V', domain)
u, v = elements_of(V, names='u, v')
c = Constant('c')
int_0 = lambda expr: integral(domain, expr)
a = BilinearForm((u, v), int_0(c * div(v) * div(u) + curl(v) * curl(u)))
expr = TensorExpr(a, mapping=M)
print(expr)
def test_compiler_3d_stokes():
domain = Domain('Omega', dim=3)
# ...
# by setting the space type, we cannot evaluate grad of Hdiv function, then
# ArgumentTypeError will be raised.
# In order to avoid this problem, we need first to declare our space as an
# undefined type.
Hdiv = VectorFunctionSpace('V2', domain, kind='Hdiv')
L2 = ScalarFunctionSpace('V3', domain, kind='L2')
X = Hdiv * L2
u, p = element_of(X, name='u, p')
v, q = element_of(X, name='v, q')
with pytest.raises(ArgumentTypeError):
expr = inner(grad(u), grad(v)) - div(v) * p + q * div(u)
# ...
# ...
Hdiv = VectorFunctionSpace('V2', domain)
L2 = ScalarFunctionSpace('V3', domain)
X = Hdiv * L2
u, p = element_of(X, name='u, p')
v, q = element_of(X, name='v, q')
expr = inner(grad(u), grad(v)) - div(v) * p + q * div(u)
atoms = {
u: DifferentialForm('u', index=2, dim=domain.dim),
v: DifferentialForm('v', index=2, dim=domain.dim),
p: DifferentialForm('p', index=3, dim=domain.dim),
q: DifferentialForm('q', index=3, dim=domain.dim)
}
newexpr = ExteriorCalculusExpr(expr, tests=[v, q], atoms=atoms)
print('===== BEFORE =====')
print(newexpr)
newexpr = augmented_expression(newexpr,
tests=[v, q],
atoms=atoms,
weak=False)
print('===== AFTER =====')
print(newexpr)
def test_compiler_3d_poisson():
domain = Domain('Omega', dim=3)
H1 = ScalarFunctionSpace('V0', domain, kind='H1')
Hcurl = VectorFunctionSpace('V1', domain, kind='Hcurl')
Hdiv = VectorFunctionSpace('V2', domain, kind='Hdiv')
L2 = ScalarFunctionSpace('V3', domain, kind='L2')
V = VectorFunctionSpace('V', domain)
X = Hdiv * L2
sigma, u = element_of(X, name='sigma, u')
tau, v = element_of(X, name='tau, v')
expr = dot(sigma, tau) + div(tau) * u + div(sigma) * v
print(ExteriorCalculusExpr(expr, tests=[tau, v]))
def test_compiler_3d_2():
domain = Domain('Omega', dim=3)
H1 = ScalarFunctionSpace('V0', domain, kind='H1')
Hcurl = VectorFunctionSpace('V1', domain, kind='Hcurl')
Hdiv = VectorFunctionSpace('V2', domain, kind='Hdiv')
L2 = ScalarFunctionSpace('V3', domain, kind='L2')
V = VectorFunctionSpace('V', domain)
X = H1 * Hcurl * Hdiv * L2
v = element_of(X, name='v0, v1, v2, v3')
u = element_of(X, name='u0, u1, u2, u3')
beta = Field(V, 'beta')
# # ... Dot operator
# expr = dot(u1, v1)
# print(ExteriorCalculusExpr(expr, tests=[v1]))
#
# expr = dot(u2, v2)
# print(ExteriorCalculusExpr(expr, tests=[v2]))
#
# expr = dot(grad(v0), u1)
# print(ExteriorCalculusExpr(expr, tests=[v0]))
#
# expr = dot(grad(u0), v1)
# print(ExteriorCalculusExpr(expr, tests=[v1]))
#
# expr = dot(curl(u1), v2)
# print(ExteriorCalculusExpr(expr, tests=[v2]))
#
# expr = dot(curl(v1), u2)
# print(ExteriorCalculusExpr(expr, tests=[v1]))
# # ...
# ... Mul operator
expr = u[0] * v[0]
print(ExteriorCalculusExpr(expr, tests=[v[0]]))
expr = u[0] * div(v[2])
print(ExteriorCalculusExpr(expr, tests=[v[2]]))
expr = v[0] * div(u[2])
print(ExteriorCalculusExpr(expr, tests=[v[0]]))
def test_tensorize_3d():
V = FunctionSpace('V', domain)
U = FunctionSpace('U', domain)
W1 = VectorFunctionSpace('W1', domain)
T1 = VectorFunctionSpace('T1', domain)
v = TestFunction(V, name='v')
u = TestFunction(U, name='u')
w1 = VectorTestFunction(W1, name='w1')
t1 = VectorTestFunction(T1, name='t1')
x,y,z = domain.coordinates
alpha = Constant('alpha')
# ...
expr = dot(grad(v), grad(u))
a = BilinearForm((v,u), expr, name='a')
print(a)
print(tensorize(a))
print('')
# ...
# ...
expr = x*dx(v)*dx(u) + y*dy(v)*dy(u)
a = BilinearForm((v,u), expr, name='a')
print(a)
print(tensorize(a))
print('')
# ...
# ...
expr = sin(x)*dx(v)*dx(u)
a = BilinearForm((v,u), expr, name='a')
print(a)
print(tensorize(a))
print('')
# ...
# ...
expr = dot(curl(w1), curl(t1)) + div(w1)*div(t1)
a = BilinearForm((w1, t1), expr, name='a')
print(a)
print(tensorize(a))
print('')
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_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_logical_expr_3d_4():
dim = 3
domain = Domain('Omega', dim=dim)
M = Mapping('M', dim=3)
mapped_domain = M(domain)
V = VectorFunctionSpace('V', domain, kind='hdiv')
VM = VectorFunctionSpace('VM', mapped_domain, kind='hdiv')
u, v = elements_of(V, names='u,v')
um, vm = elements_of(VM, names='u,v')
J = M.jacobian
a = div(um) * div(vm)
e = LogicalExpr(a, mapping=M, dim=dim)
assert e == J.det()**-2 * div(u) * div(v)
def test_tensorize_2d_2():
domain = Domain('Omega', dim=2)
B1 = Boundary(r'\Gamma_1', domain)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
V = VectorFunctionSpace('V', domain)
u, v = elements_of(V, names='u, v')
int_0 = lambda expr: integral(domain, expr)
# ...
# a = BilinearForm((u,v), dot(u,v))
a = BilinearForm((u, v), int_0(curl(u) * curl(v) + div(u) * div(v)))
expr = TensorExpr(a)
print(expr)
def test_bilinear_form_2d_3():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
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((u, v), inner(grad(v), grad(u)))
b = BilinearForm((v, p), div(v) * p)
A = BilinearForm(((u, p), (v, q)), a(v, u) - b(v, p) + b(u, q))
export(A, 'stokes_2d.png')
def test_tensorize_2d_stokes():
domain = Domain('Omega', dim=DIM)
# ... abstract model
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(tensorize(A))
print('')
def test_terminal_expr_bilinear_2d_4():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
V = VectorFunctionSpace('V', domain)
W = ScalarFunctionSpace('W', domain)
v = element_of(V, name='v')
u = element_of(V, name='u')
p = element_of(W, name='p')
q = element_of(W, name='q')
int_0 = lambda expr: integral(domain, expr)
# stokes
a = BilinearForm((u, v), int_0(inner(grad(v), grad(u))))
b = BilinearForm((v, p), int_0(div(v) * p))
a = BilinearForm(((u, p), (v, q)), a(v, u) - b(v, p) + b(u, q))
print(TerminalExpr(a))
print('')
def test_latex_2d_4():
DIM = 2
domain = Domain('Omega', dim=DIM)
# ... abstract model
V = VectorFunctionSpace('V', domain)
W = ScalarFunctionSpace('W', domain)
v = element_of(V, name='v')
u = element_of(V, name='u')
p = element_of(W, name='p')
q = element_of(W, name='q')
int_0 = lambda expr: integral(domain, expr)
a = BilinearForm((v, u), int_0(inner(grad(v), grad(u))))
b = BilinearForm((v, p), int_0(div(v) * p))
A = BilinearForm(((v, q), (u, p)), a(v, u) - b(v, p) + b(u, q))
# ...
print(latex(A))
# print(latex(tensorize(A)))
print('')
def test_calculus_2d_5():
DIM = 2
domain = Domain('Omega', dim=DIM)
V = VectorFunctionSpace('V', domain, kind=None)
u, v = elements_of(V, names='u, v')
a = Constant('a', is_real=True)
# ... jump operator
assert (jump(u + v) == jump(u) + jump(v))
assert (jump(a * u) == a * jump(u))
# ...
# ... avg operator
assert (avg(u + v) == avg(u) + avg(v))
assert (avg(a * u) == a * avg(u))
# ...
# ... Dn operator
assert (Dn(u + v) == Dn(u) + Dn(v))
assert (Dn(a * u) == a * Dn(u))
# ...
# ... minus operator
assert (minus(u + v) == minus(u) + minus(v))
assert (minus(a * u) == a * minus(u))
assert (div(minus(u + v)) == div(minus(u)) + div(minus(v)))
# ...
# ... plus operator
assert (plus(u + v) == plus(u) + plus(v))
assert (plus(a * u) == a * plus(u))
assert (div(plus(u + v)) == div(plus(u)) + div(plus(v)))
def test_bilinearity_2d_2():
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, check=True)
a = BilinearForm((v2, u2), a1(v2, u2), check=True)
# ...
# ...
a = BilinearForm((v1, u1), dot(grad(v1), grad(u1)), check=True)
# ...
# ...
a1 = BilinearForm((v1, u1), dot(grad(v1), grad(u1)), check=True)
a = BilinearForm((v2, u2), a1(v2, u2), check=True)
# ...
# ...
a1 = BilinearForm((v1, u1), u1 * v1)
a2 = BilinearForm((v1, u1), dx(u1) * dx(v1), check=True)
a = BilinearForm((v2, u2), a1(v2, u2) + a2(v2, u2), check=True)
# ...
# ...
a1 = BilinearForm((v1, u1), u1 * v1, check=True)
a2 = BilinearForm((v1, u1), dx(u1) * dx(v1), check=True)
# ...
# ...
a1 = BilinearForm((v1, u1), u1 * v1, check=True)
a2 = BilinearForm((v1, u1), dx(u1) * dx(v1), check=True)
a = BilinearForm(((v1, v2), (u1, u2)), a1(v1, u2) + a2(v2, u1), check=True)
# ...
# ...
a = BilinearForm((w1, t1),
rot(w1) * rot(t1) + div(w1) * div(t1),
check=True)
# ...
# ...
a1 = BilinearForm((v1, u1), u1 * v1, check=True)
a2 = BilinearForm((v1, u1), dx(u1) * dx(v1), check=True)
a3 = BilinearForm((w1, t1),
rot(w1) * rot(t1) + div(w1) * div(t1),
check=True)
a4 = BilinearForm((w1, u1), div(w1) * u1, check=True)
a = BilinearForm(((w2, v2), (t2, u2)),
a3(w2, t2) + a2(v2, u2) + a4(w2, u2),
check=True)
# ...
# ...
a1 = BilinearForm((v1, u1), laplace(u1) * laplace(v1), check=True)
# ...
# ...
a1 = BilinearForm((v1, u1), inner(hessian(u1), hessian(v1)), check=True)
# ...
# ... 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)), check=True)
b = BilinearForm((v, p), div(v) * p, check=True)
A = BilinearForm(((v, q), (u, p)), a(v, u) - b(v, p) + b(u, q), check=True)
# ...
################################
# non bilinear forms
################################
# ...
with pytest.raises(UnconsistentLinearExpressionError):
a = BilinearForm((v1, u1), dot(grad(v1), grad(u1)) + v1, check=True)
# ...
# ...
with pytest.raises(UnconsistentLinearExpressionError):
a = BilinearForm((v1, u1), v1**2 * u1, check=True)
# ...
# ...
with pytest.raises(UnconsistentLinearExpressionError):
a = BilinearForm((v1, u1), dot(grad(v1), grad(v1)), check=True)
def test_linearity_2d_2():
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)
# ...
l1 = LinearForm(v1, x * y * v1, check=True)
l = LinearForm(v2, l1(v2), check=True)
# ...
# ...
l1 = LinearForm(v1, x * y * v1, check=True)
l2 = LinearForm(v2, cos(x + y) * v2, check=True)
l = LinearForm((u1, u2), l1(u1) + l2(u2), check=True)
# ...
# ...
l1 = LinearForm(v1, x * y * v1, check=True)
l2 = LinearForm(v2, cos(x + y) * v2, check=True)
l = LinearForm((u1, u2), l1(u1) + alpha * l2(u2), check=True)
# ...
# ...
l1 = LinearForm(v1, x * y * v1, check=True)
l2 = LinearForm(w1, div(w1), check=True)
l = LinearForm((v2, w2), l1(v2) + l2(w2), check=True)
# ...
################################
# non bilinear forms
################################
# ...
with pytest.raises(UnconsistentLinearExpressionError):
l = LinearForm(v1, x * y * v1**2, check=True)
# ...
# ...
with pytest.raises(UnconsistentLinearExpressionError):
l = LinearForm(v1, x * y, check=True)
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_compiler_3d_1():
domain = Domain('Omega', dim=3)
H1 = ScalarFunctionSpace('V0', domain, kind='H1')
Hcurl = VectorFunctionSpace('V1', domain, kind='Hcurl')
Hdiv = VectorFunctionSpace('V2', domain, kind='Hdiv')
L2 = ScalarFunctionSpace('V3', domain, kind='L2')
V = VectorFunctionSpace('V', domain)
X = H1 * Hcurl * Hdiv * L2
v0, v1, v2, v3 = element_of(X, name='v0, v1, v2, v3')
beta = element_of(V, 'beta')
# # ...
# expr = grad(v0)
# expected = d(DifferentialForm('v0', index=0, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr) == expected)
# # ...
#
# # ...
# expr = curl(v1)
# expected = d(DifferentialForm('v1', index=1, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr) == expected)
# # ...
#
# # ...
# expr = div(v2)
# expected = d(DifferentialForm('v2', index=2, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr) == expected)
# # ...
#
# # ...
# expr = grad(v0)
# expected = - delta(DifferentialForm('v0', index=3, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr, tests=[v0]) == expected)
# # ...
#
# # ...
# expr = curl(v1)
# expected = delta(DifferentialForm('v1', index=2, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr, tests=[v1]) == expected)
# # ...
#
# # ...
# expr = div(v2)
# expected = -delta(DifferentialForm('v2', index=1, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr, tests=[v2]) == expected)
# # ...
#
# # ...
# expr = dot(beta, v1)
# expected = ip(beta, DifferentialForm('v1', index=1, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr) == expected)
# # ...
#
# # ...
# expr = cross(beta, v2)
# expected = ip(beta, DifferentialForm('v2', index=2, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr) == expected)
# # ...
#
# # ...
# expr = beta*v3
# expected = ip(beta, DifferentialForm('v3', index=3, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr) == expected)
# # ...
#
# # ...
# expr = dot(beta, v1)
# expected = jp(beta, DifferentialForm('v1', index=3, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr, tests=[v1]) == expected)
# # ...
#
# # ...
# expr = cross(beta, v2)
# expected = -jp(beta, DifferentialForm('v2', index=2, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr, tests=[v2]) == expected)
# # ...
#
# # ...
# expr = beta*v3
# expected = jp(beta, DifferentialForm('v3', index=1, dim=domain.dim))
#
# assert(ExteriorCalculusExpr(expr, tests=[v3]) == expected)
# # ...
# ..................................................
# LIE DERIVATIVES
# ..................................................
# ...
expr = dot(beta, grad(v0))
expected = ld(beta, DifferentialForm('v0', index=0, dim=domain.dim))
assert (ExteriorCalculusExpr(expr) == expected)
# ...
# ...
expr = grad(dot(beta, v1)) + cross(curl(v1), beta)
expected = ld(beta, DifferentialForm('v1', index=1, dim=domain.dim))
assert (ExteriorCalculusExpr(expr) == expected)
# ...
# ...
expr = curl(cross(v2, beta)) + div(v2) * beta
expected = ld(beta, DifferentialForm('v2', index=2, dim=domain.dim))
#assert(ExteriorCalculusExpr(expr) == expected)
# ...
# ...
expr = div(beta * v3)
expected = ld(beta, DifferentialForm('v3', index=3, dim=domain.dim))
assert (ExteriorCalculusExpr(expr) == expected)
def test_calculus_3d_3():
domain = Domain('Omega', dim=3)
H1 = ScalarFunctionSpace('V0', domain, kind='H1')
Hcurl = VectorFunctionSpace('V1', domain, kind='Hcurl')
Hdiv = VectorFunctionSpace('V2', domain, kind='Hdiv')
L2 = ScalarFunctionSpace('V3', domain, kind='L2')
V = ScalarFunctionSpace('V', domain, kind=None)
W = VectorFunctionSpace('W', domain, kind=None)
X = ProductSpace(H1, Hcurl, Hdiv, L2)
v0, v1, v2, v3 = element_of(X, ['v0', 'v1', 'v2', 'v3'])
v = element_of(V, 'v')
w = element_of(W, 'w')
# ... consistency of grad operator
# we can apply grad on an undefined space type or H1
expr = grad(v0)
expr = grad(v)
expr = grad(w)
# wa cannot apply grad to a function in Hcurl
with pytest.raises(ArgumentTypeError):
expr = grad(v1)
# wa cannot apply grad to a function in Hdiv
with pytest.raises(ArgumentTypeError):
expr = grad(v2)
# wa cannot apply grad to a function in L2
with pytest.raises(ArgumentTypeError):
expr = grad(v3)
# ...
# ... consistency of curl operator
# we can apply curl on an undefined space type, H1 or Hcurl
expr = curl(v0)
expr = curl(v1)
expr = curl(v)
expr = curl(w)
# wa cannot apply curl to a function in Hdiv
with pytest.raises(ArgumentTypeError):
expr = curl(v2)
# wa cannot apply curl to a function in L2
with pytest.raises(ArgumentTypeError):
expr = curl(v3)
# ...
# ... consistency of div operator
# we can apply div on an undefined space type, H1 or Hdiv
expr = div(v0)
expr = div(v2)
expr = div(v)
expr = div(w)
# wa cannot apply div to a function in Hcurl
with pytest.raises(ArgumentTypeError):
expr = div(v1)
# wa cannot apply div to a function in L2
with pytest.raises(ArgumentTypeError):
expr = div(v3)
