def test_logical_expr_3d_5():
dim = 3
domain = Domain('Omega', dim=dim)
M = Mapping('M', dim=dim)
mapped_domain = M(domain)
V = VectorFunctionSpace('V', domain, kind='hcurl')
VM = VectorFunctionSpace('VM', mapped_domain, kind='hcurl')
J = M.jacobian
u, v = elements_of(V, names='u,v')
um, vm = elements_of(VM, names='u,v')
int_md = lambda expr: integral(mapped_domain, expr)
int_ld = lambda expr: integral(domain, expr)
am = BilinearForm((um, vm), int_md(dot(curl(vm), curl(um))))
a = LogicalExpr(am)
assert a == BilinearForm(
(u, v),
int_ld(
sqrt((J.T * J).det()) *
dot(J / J.det() * curl(u), J / J.det() * curl(v))))
def test_zero_derivative():
assert grad(1) == 0 # native int
assert grad(2.3) == 0 # native float
assert grad(4 + 5j) == 0 # native complex
assert grad(Integer(1)) == 0 # sympy Integer
assert grad(Float(2.3)) == 0 # sympy Float
assert grad(Rational(6, 7)) == 0 # sympy Rational
assert grad(Constant('a')) == 0 # sympde Constant
assert laplace(1) == 0 # native int
assert laplace(2.3) == 0 # native float
assert laplace(4 + 5j) == 0 # native complex
assert laplace(Integer(1)) == 0 # sympy Integer
assert laplace(Float(2.3)) == 0 # sympy Float
assert laplace(Rational(6, 7)) == 0 # sympy Rational
assert laplace(Constant('a')) == 0 # sympde Constant
assert hessian(1) == 0 # native int
assert hessian(2.3) == 0 # native float
assert hessian(4 + 5j) == 0 # native complex
assert hessian(Integer(1)) == 0 # sympy Integer
assert hessian(Float(2.3)) == 0 # sympy Float
assert hessian(Rational(6, 7)) == 0 # sympy Rational
assert hessian(Constant('a')) == 0 # sympde Constant
# 2D convection of constant scalar field
domain = Domain('Omega', dim=2)
W = VectorFunctionSpace('W', domain)
F = element_of(W, name='F')
assert convect(F, 1) == 0 # native int
assert convect(F, 2.3) == 0 # native float
assert convect(F, 4 + 5j) == 0 # native complex
assert convect(F, Integer(1)) == 0 # sympy Integer
assert convect(F, Float(2.3)) == 0 # sympy Float
assert convect(F, Rational(6, 7)) == 0 # sympy Rational
assert convect(F, Constant('a')) == 0 # sympde Constant
# 3D convection of constant scalar field
domain = Domain('Omega', dim=3)
Z = VectorFunctionSpace('Z', domain)
G = element_of(Z, name='G')
assert convect(G, 1) == 0 # native int
assert convect(G, 2.3) == 0 # native float
assert convect(G, 4 + 5j) == 0 # native complex
assert convect(G, Integer(1)) == 0 # sympy Integer
assert convect(G, Float(2.3)) == 0 # sympy Float
assert convect(G, Rational(6, 7)) == 0 # sympy Rational
assert convect(G, Constant('a')) == 0 # sympde Constant
def test_tensorize_2d():
domain = Domain('Omega', dim=DIM)
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 = 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 = rot(w1)*rot(t1) + div(w1)*div(t1)
expr = rot(w1) * rot(t1) #+ div(w1)*div(t1)
a = BilinearForm((w1, t1), expr, name='a')
print(a)
print(tensorize(a))
print('')
def test_linear_expr_2d_2():
domain = Domain('Omega', dim=2)
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']]
g = Tuple(x, y)
l = LinearExpr(v, dot(g, v))
print(l)
print(l.expr)
print(l(v1))
# TODO
# print(l(v1+v2))
print('')
# ...
# ...
g1 = Tuple(x, 0)
g2 = Tuple(0, y)
l = LinearExpr((v1, v2), dot(g1, v1) + dot(g2, v2))
print(l)
print(l.expr)
print(l(u1, u2))
# TODO
# print(l(u1+v1, u2+v2))
print('')
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_bilinear_expr_2d_2():
domain = Domain('Omega', dim=2)
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']]
# ...
a = BilinearExpr((u,v), dot(u,v))
print(a)
print(a.expr)
print(a(u1,v1))
# TODO
# print(a(u1+u2,v1+v2))
print('')
# ...
# ...
a1 = BilinearExpr((u,v), dot(u,v))
a2 = BilinearExpr((u,v), inner(grad(u),grad(v)))
print(a1(u1,v1) + a2(u2,v2))
print('')
def test_equation_2d_4():
V = VectorFunctionSpace('V', domain)
v = element_of(V, name='v')
u = element_of(V, name='u')
x, y = domain.coordinates
B1 = Boundary(r'\Gamma_1', domain)
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
# ... bilinear/linear forms
a1 = BilinearForm((v, u), int_0(inner(grad(v), grad(u))))
f = Tuple(x * y, sin(pi * x) * sin(pi * y))
l1 = LinearForm(v, int_0(dot(f, v)))
# ...
# ...
bc = EssentialBC(u, 0, B1)
eq = Equation(a1, l1, tests=v, trials=u, bc=bc)
# ...
# ...
bc = EssentialBC(u[0], 0, B1)
eq = Equation(a1, l1, tests=v, trials=u, bc=bc)
# ...
# ...
nn = NormalVector('nn')
bc = EssentialBC(dot(u, nn), 0, B1)
eq = Equation(a1, l1, tests=v, trials=u, bc=bc)
def test_Inner(dim):
domain = Domain('Omega', dim=dim)
W = VectorFunctionSpace('W', domain)
a, b, c = elements_of(W, names='a, b, c')
r = Constant('r')
# Commutativity
assert inner(a, b) == inner(b, a)
# Bilinearity: vector addition
assert inner(a, b + c) == inner(a, b) + inner(a, c)
assert inner(a + b, c) == inner(a, c) + inner(b, c)
# Bilinearity: scalar multiplication
assert inner(a, r * b) == r * inner(a, b)
assert inner(r * a, b) == r * inner(a, b)
# Special case: null vector
assert inner(a, 0) == 0
assert inner(0, a) == 0
# Special case: two arguments are the same
assert inner(a, a).is_real
assert inner(a, a).is_positive
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_terminal_expr_bilinear_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)
nn = NormalVector('nn')
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)
a = BilinearForm((u, v), int_0(dot(u, v)))
print(TerminalExpr(a))
print('')
# ...
a = BilinearForm((u, v), int_0(inner(grad(u), grad(v))))
print(TerminalExpr(a))
print('')
# ...
# ...
a = BilinearForm((u, v), int_0(dot(u, v) + inner(grad(u), grad(v))))
print(TerminalExpr(a))
print('')
def test_bilinear_form_2d_4():
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)
# ...
a = BilinearForm((u, v), int_0(dot(u, v)))
assert (a.is_symmetric)
# ...
# ...
a = BilinearForm((u, v), int_0(inner(grad(u), grad(v))))
assert (a.is_symmetric)
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_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_ec_3d_1():
n = 3
# ...
u_0 = DifferentialForm('u_0', index=0, dim=n)
v_0 = DifferentialForm('v_0', index=0, dim=n)
u_1 = DifferentialForm('u_1', index=1, dim=n)
v_1 = DifferentialForm('v_1', index=1, dim=n)
u_2 = DifferentialForm('u_2', index=2, dim=n)
v_2 = DifferentialForm('v_2', index=2, dim=n)
u_3 = DifferentialForm('u_3', index=3, dim=n)
v_3 = DifferentialForm('v_3', index=3, dim=n)
# ...
# ...
domain = Domain('Omega', dim=3)
V = VectorFunctionSpace('V', domain)
beta = element_of(V, 'beta')
# ...
print(latex(u_0))
print(latex(d(u_0)))
print(latex(d(delta(u_3))))
print(latex(d(delta(u_2)) + delta(d(u_2))))
print(latex(wedge(u_0, u_1)))
print(latex(ip(beta, u_1)))
print(latex(hodge(u_1)))
print(latex(jp(beta, u_1)))
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_logical_expr_2d_3():
dim = 2
A = Square('A')
B = Square('B')
M1 = Mapping('M1', dim=dim)
M2 = Mapping('M2', dim=dim)
D1 = M1(A)
D2 = M2(B)
domain = D1.join(D2,
name='domain',
bnd_minus=D1.get_boundary(axis=0, ext=1),
bnd_plus=D2.get_boundary(axis=0, ext=-1))
V = VectorFunctionSpace('V', domain, kind='hcurl')
u, v = [element_of(V, name=i) for i in ['u', 'v']]
int_0 = lambda expr: integral(domain, expr)
expr = LogicalExpr(int_0(dot(u, v)), domain)
assert str(
expr.args[0]
) == 'Integral(A, Dot((Jacobian(M1)**(-1)).T * u, (Jacobian(M1)**(-1)).T * v)*sqrt(det(Jacobian(M1).T * Jacobian(M1))))'
assert str(
expr.args[1]
) == 'Integral(B, Dot((Jacobian(M2)**(-1)).T * u, (Jacobian(M2)**(-1)).T * v)*sqrt(det(Jacobian(M2).T * Jacobian(M2))))'
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_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_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_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_logical_expr_2d_1():
rdim = 2
M = Mapping('M', rdim)
domain = M(Domain('Omega', dim=rdim))
alpha = Constant('alpha')
V = ScalarFunctionSpace('V', domain, kind='h1')
W = VectorFunctionSpace('V', domain, kind='h1')
u, v = [element_of(V, name=i) for i in ['u', 'v']]
w = element_of(W, name='w')
det_M = Jacobian(M).det()
#print('det = ', det_M)
det = Symbol('det')
# ...
expr = 2 * u + alpha * v
expr = LogicalExpr(expr, mapping=M, dim=rdim)
#print(expr)
#print('')
# ...
# ...
expr = dx(u)
expr = LogicalExpr(expr, mapping=M, dim=rdim)
#print(expr.subs(det_M, det))
#print('')
# ...
# ...
expr = dy(u)
expr = LogicalExpr(expr, mapping=M, dim=rdim)
#print(expr.subs(det_M, det))
#print('')
# ...
# ...
expr = dx(det_M)
expr = LogicalExpr(expr, mapping=M, dim=rdim)
expr = expr.subs(det_M, det)
expr = expand(expr)
#print(expr)
#print('')
# ...
# ...
expr = dx(dx(u))
expr = LogicalExpr(expr, mapping=M, dim=rdim)
#print(expr.subs(det_M, det))
#print('')
# ...
# ...
expr = dx(w[0])
expr = LogicalExpr(expr, mapping=M, dim=rdim)
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_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_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_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_linearize_2d_1():
domain = Domain('Omega', dim=DIM)
x, y = domain.coordinates
V1 = FunctionSpace('V1', domain)
W1 = VectorFunctionSpace('W1', domain)
v1 = TestFunction(V1, name='v1')
w1 = VectorTestFunction(W1, name='w1')
alpha = Constant('alpha')
F = Field('F', space=V1)
G = VectorField(W1, 'G')
# ...
l = LinearForm(v1, F**2 * v1, check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(v1, dot(grad(F), grad(F)) * v1, check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(v1, exp(-F) * v1, check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(v1, cos(F) * v1, check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(v1, cos(F**2) * v1, check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(v1, F**2 * dot(grad(F), grad(v1)), check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(w1, dot(rot(G), grad(G)) * w1, check=True)
a = linearize(l, G, trials='u1')
print(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_logical_expr_3d_3():
dim = 3
domain = Domain('Omega', dim=dim)
M = Mapping('M', dim=3)
mapped_domain = M(domain)
V = VectorFunctionSpace('V', domain, kind='hcurl')
VM = VectorFunctionSpace('VM', mapped_domain, kind='hcurl')
u, v = elements_of(V, names='u,v')
um, vm = elements_of(VM, names='u,v')
J = M.jacobian
a = dot(curl(um), curl(vm))
e = LogicalExpr(a, mapping=M, dim=dim)
assert e == dot(J / J.det() * curl(u), J / J.det() * curl(v))
def test_calculus_3d_4():
domain = Domain('Omega', dim=3)
W = VectorFunctionSpace('W', domain)
alpha, beta, gamma = [Constant(i) for i in ['alpha', 'beta', 'gamma']]
F, G, H = elements_of(W, names='F, G, H')
# ...
expected = alpha * inner(D(F), D(G)) + beta * inner(D(F), D(H))
assert (inner(D(F), D(alpha * G + beta * H)) == expected)
def test_calculus_3d_5():
domain = Domain('Omega', dim=3)
W = VectorFunctionSpace('W', domain)
alpha, beta, gamma = [Constant(i) for i in ['alpha', 'beta', 'gamma']]
F, G, H = elements_of(W, names='F, G, H')
# ...
expected = alpha * outer(F, G) + beta * outer(F, H)
assert (outer(F, alpha * G + beta * H) == expected)