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_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_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_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 = ScalarFunctionSpace('V', domain)
u,v = [element_of(V, name=i) for i in ['u', 'v']]
int_0 = lambda expr: integral(domain , expr)
# ...
expr = dot(grad(u), grad(v)) + f(x,y) * u * v
a = BilinearForm((v,u), int_0(expr))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
expr = f(x,y) * v
l = LinearForm(v, int_0(expr))
print(l)
print(TerminalExpr(l))
print('')
def test_functional_2d_1():
domain = Domain('Omega', dim=2)
x,y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu' , is_real=True)
V = ScalarFunctionSpace('V', domain)
F = element_of(V, name='F')
int_0 = lambda expr: integral(domain , expr)
# ...
expr = x*y
a = Functional(int_0(expr), domain)
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
expr = F - cos(2*pi*x)*cos(3*pi*y)
expr = dot(grad(expr), grad(expr))
a = Functional(int_0(expr), domain)
print(a)
print(TerminalExpr(a))
print('')
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 = Tuple(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_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_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 = Matrix([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_equation_2d_3():
V = ScalarFunctionSpace('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(dot(grad(v), grad(u))))
a2 = BilinearForm((v, u), int_0(v * u))
l1 = LinearForm(v, int_0(x * y * v))
l2 = LinearForm(v, int_0(cos(x + y) * v))
# ...
# ...
bc = EssentialBC(u, 0, B1)
eq = Equation(a1, l1, tests=v, trials=u, bc=bc)
# ...
# ...
nn = NormalVector('nn')
bc = EssentialBC(dot(grad(u), nn), 0, B1)
eq = Equation(a1, l1, tests=v, trials=u, bc=bc)
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_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_interface_2d_1():
# ...
def two_patches():
from sympde.topology import InteriorDomain
from sympde.topology import Connectivity, Interface
A = Square('A')
B = Square('B')
A = A.interior
B = B.interior
connectivity = Connectivity()
bnd_A_1 = Boundary(r'\Gamma_1', A, axis=0, ext=-1)
bnd_A_2 = Boundary(r'\Gamma_2', A, axis=0, ext=1)
bnd_A_3 = Boundary(r'\Gamma_3', A, axis=1, ext=-1)
bnd_A_4 = Boundary(r'\Gamma_4', A, axis=1, ext=1)
bnd_B_1 = Boundary(r'\Gamma_1', B, axis=0, ext=-1)
bnd_B_2 = Boundary(r'\Gamma_2', B, axis=0, ext=1)
bnd_B_3 = Boundary(r'\Gamma_3', B, axis=1, ext=-1)
bnd_B_4 = Boundary(r'\Gamma_4', B, axis=1, ext=1)
connectivity['I'] = Interface('I', bnd_A_2, bnd_B_1)
Omega = Domain('Omega',
interiors=[A, B],
boundaries=[
bnd_A_1, bnd_A_2, bnd_A_3, bnd_A_4, bnd_B_1,
bnd_B_2, bnd_B_3, bnd_B_4
],
connectivity=connectivity)
return Omega
# ...
# create a domain with an interface
domain = two_patches()
interfaces = domain.interfaces
V = ScalarFunctionSpace('V', domain)
u, v = elements_of(V, names='u, v')
print(integral(interfaces, u * v))
expr = integral(domain, dot(grad(v), grad(u)))
expr += integral(interfaces, -avg(Dn(u)) * jump(v) + avg(Dn(v)) * jump(u))
a = BilinearForm((u, v), expr)
print(a)
def test_interface_integral_1():
# ...
A = Square('A')
B = Square('B')
domain = A.join(B,
name='domain',
bnd_minus=A.get_boundary(axis=0, ext=1),
bnd_plus=B.get_boundary(axis=0, ext=-1))
# ...
x, y = domain.coordinates
V = ScalarFunctionSpace('V', domain, kind=None)
assert (V.is_broken)
u, v = elements_of(V, names='u, v')
# ...
I = domain.interfaces
# ...
# expr = minus(Dn(u))
# print(expr)
# import sys; sys.exit(0)
# ... bilinear forms
# a = BilinearForm((u,v), integral(domain, u*v))
# a = BilinearForm((u,v), integral(domain, dot(grad(u),grad(v))))
# a = BilinearForm((u,v), integral(I, jump(u) * jump(v)))
# a = BilinearForm((u,v), integral(I, jump(Dn(u)) * jump(v)))
# a = BilinearForm((u,v), integral(domain, dot(grad(u),grad(v)))
# + integral(I, jump(u) * jump(v)))
# Nitsch
kappa = Constant('kappa')
expr_I = (-jump(u) * jump(Dn(v)) + kappa * jump(u) * jump(v) +
plus(Dn(u)) * minus(v) + minus(Dn(u)) * plus(v))
a = BilinearForm(
(u, v),
integral(domain, dot(grad(u), grad(v))) + integral(I, expr_I))
# # TODO BUG
# bnd_A = A.get_boundary(axis=0, ext=1)
#
# a = BilinearForm((u,v), integral(domain, dot(grad(u),grad(v)))
# + integral(I, jump(u) * jump(v))
# + integral(bnd_A, dx(u)*v))
expr = TerminalExpr(a)
print(expr)
def test_interface_integral_3():
# ...
A = Square('A')
B = Square('B')
C = Square('C')
AB = A.join(B,
name='AB',
bnd_minus=A.get_boundary(axis=0, ext=1),
bnd_plus=B.get_boundary(axis=0, ext=-1))
domain = AB.join(C,
name='domain',
bnd_minus=B.get_boundary(axis=0, ext=1),
bnd_plus=C.get_boundary(axis=0, ext=-1))
# ...
x, y = domain.coordinates
V = ScalarFunctionSpace('V', domain, kind=None)
assert (V.is_broken)
u, v = elements_of(V, names='u, v')
# ...
I = domain.interfaces
# print(I)
# print(integral(I, jump(u) * jump(v)))
# a = BilinearForm((u,v), integral(domain, u*v))
# a = BilinearForm((u,v), integral(domain, dot(grad(u),grad(v))))
# a = BilinearForm((u,v), integral(I, jump(u) * jump(v)))
a = BilinearForm((u, v),
integral(domain, dot(grad(u), grad(v))) +
integral(I,
jump(u) * jump(v)))
expr = TerminalExpr(a)
print(expr)
# ...
# ... linear forms
b = LinearForm(
v,
integral(domain,
sin(x + y) * v) + integral(I,
cos(x + y) * jump(v)))
expr = TerminalExpr(b)
print(expr)
def test_field_2d_1():
print('============ test_field_2d_1 =============')
# x, y = domain.coordinates
W = VectorFunctionSpace('W', domain)
F = element_of(W, 'F')
assert (dx(F) == Matrix([[dx(F[0]), dx(F[1])]]))
# TODO not working yet => check it for VectorFunction also
# print(dx(x*F))
expr = inner(grad(F), grad(F))
print(expr)
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_equation_2d_5():
domain = Square()
x, y = domain.coordinates
f0 = Matrix([
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 = ScalarFunctionSpace('V', domain)
X = ProductSpace(W, V)
F = element_of(W, name='F')
G = element_of(V, name='G')
u, v = [element_of(W, name=i) for i in ['u', 'v']]
p, q = [element_of(V, name=i) for i in ['p', 'q']]
int_0 = lambda expr: integral(domain, expr)
a0 = BilinearForm((v, u), int_0(inner(grad(v), grad(u))))
print(' a0 done.')
a1 = BilinearForm((q, p), int_0(p * q))
print(' a1 done.')
a = BilinearForm(((v, q), (u, p)), a0(v, u) + a1(q, p))
print(' a done.')
l0 = LinearForm(v, int_0(dot(f0, v)))
l1 = LinearForm(q, int_0(f1 * q))
l = LinearForm((v, q), l0(v) + l1(q))
print('****************************')
bc = EssentialBC(u, 0, domain.boundary)
equation = Equation(a, l, tests=[v, q], trials=[u, p], bc=bc)
# ...
print('=======')
print(equation.lhs.expr)
print('')
# ...
# ...
print('=======')
print(equation.rhs.expr)
print('')
def test_tensorize_2d_3():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
u, v = elements_of(V, names='u,v')
bx = Constant('bx')
by = Constant('by')
b = Tuple(bx, by)
expr = integral(domain, dot(b, grad(v)) * dot(b, grad(u)))
a = BilinearForm((u, v), expr)
print(TensorExpr(a))
print('')
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_expr_mapping_2d():
F = Mapping('F', DIM)
patch = Domain('Omega', dim=DIM)
domain = F(patch)
V = FunctionSpace('V', domain)
v = TestFunction(V, name='v')
u = TestFunction(V, name='u')
x, y = V.coordinates
a = BilinearForm((v, u), dot(grad(v), grad(u)))
assert (a.mapping is F)
l = LinearForm(v, x * y * v)
assert (l.mapping is F)
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_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_terminal_expr_linear_2d_5(boundary=[r'\Gamma_1', r'\Gamma_3']):
# ... abstract model
domain = Square()
V = ScalarFunctionSpace('V', domain)
B_neumann = [domain.get_boundary(i) for i in boundary]
if len(B_neumann) == 1:
B_neumann = B_neumann[0]
else:
B_neumann = Union(*B_neumann)
x, y = domain.coordinates
nn = NormalVector('nn')
F = element_of(V, name='F')
v = element_of(V, name='v')
u = element_of(V, name='u')
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B_neumann, expr)
expr = dot(grad(v), grad(u))
a = BilinearForm((v, u), int_0(expr))
solution = cos(0.5 * pi * x) * cos(0.5 * pi * y)
f = (1. / 2.) * pi**2 * solution
expr = f * v
l0 = LinearForm(v, int_0(expr))
expr = v * dot(grad(solution), nn)
l_B_neumann = LinearForm(v, int_1(expr))
expr = l0(v) + l_B_neumann(v)
l = LinearForm(v, expr)
print(TerminalExpr(l))
print('')
def test_interface_integral_2():
# ...
A = Square('A')
B = Square('B')
domain = A.join(B,
name='domain',
bnd_minus=A.get_boundary(axis=0, ext=1),
bnd_plus=B.get_boundary(axis=0, ext=-1))
# ...
x, y = domain.coordinates
V = ScalarFunctionSpace('V', domain, kind=None)
assert (V.is_broken)
u, u1, u2, u3 = elements_of(V, names='u, u1, u2, u3')
v, v1, v2, v3 = elements_of(V, names='v, v1, v2, v3')
# ...
I = domain.interfaces
a = BilinearForm((u, v), integral(domain, dot(grad(u), grad(v))))
b = BilinearForm((u, v), integral(I, jump(u) * jump(v)))
A = BilinearForm(((u1, u2), (v1, v2)),
a(u1, v1) + a(u2, v2) + b(u1, v1) + b(u2, v2) + b(u1, v2))
B = BilinearForm(
((u1, u2, u3), (v1, v2, v3)),
a(u1, v1) + a(u2, v2) + a(u3, v3) + b(u1, v1) + b(u2, v2) + b(u1, v2))
print(TerminalExpr(A))
print(TerminalExpr(B))
# ...
# ... linear forms
b = LinearForm(v, integral(I, jump(v)))
b = LinearForm((v1, v2), b(v1) + b(v2))
expr = TerminalExpr(b)
print(expr)
def test_tensorize_2d_1():
domain = Domain('Omega', dim=2)
mu = Constant('mu' , is_real=True)
V = ScalarFunctionSpace('V', domain)
u, v = elements_of(V, names='u, v')
int_0 = lambda expr: integral(domain , expr)
# ...
# a = BilinearForm((u,v), u*v)
a = BilinearForm((u,v), int_0(mu*u*v + dot(grad(u),grad(v))))
# a = BilinearForm((u,v), dot(grad(u),grad(v)))
# a = BilinearForm((u,v), dx(u)*v)
# a = BilinearForm((u,v), laplace(u)*laplace(v))
expr = TensorExpr(a, domain=domain)
print(expr)
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_logical_expr_3d_2():
dim = 3
domain = Domain('Omega', dim=dim)
M = Mapping('M', dim=dim)
mapped_domain = M(domain)
V = ScalarFunctionSpace('V', domain, kind='h1')
VM = ScalarFunctionSpace('VM', mapped_domain, kind='h1')
u, v = elements_of(V, names='u,v')
um, vm = elements_of(VM, names='u,v')
J = M.jacobian
a = dot(grad(um), grad(vm))
e = LogicalExpr(a, mapping=M, dim=dim)
assert e == dot(J.inv().T * grad(u), J.inv().T * grad(v))
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_essential_bc_1():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
W = VectorFunctionSpace('W', domain)
v = element_of(V, name='v')
w = element_of(W, name='w')
B1 = Boundary(r'\Gamma_1', domain)
nn = NormalVector('nn')
# ... scalar case
bc = EssentialBC(v, 0, B1)
assert (bc.variable == v)
assert (bc.order == 0)
assert (bc.normal_component == False)
assert (bc.index_component == None)
# ...
# ... scalar case
bc = EssentialBC(dot(grad(v), nn), 0, B1)
assert (bc.variable == v)
assert (bc.order == 1)
assert (bc.normal_component == False)
assert (bc.index_component == None)
# ...
# ... vector case
bc = EssentialBC(w, 0, B1)
assert (bc.variable == w)
assert (bc.order == 0)
assert (bc.normal_component == False)
assert (bc.index_component == [0, 1])
# ...
# ... vector case
bc = EssentialBC(dot(w, nn), 0, B1)
assert (bc.variable == w)
assert (bc.order == 0)
assert (bc.normal_component == True)
assert (bc.index_component == None)
# ...
# ... vector case
bc = EssentialBC(w[0], 0, B1)
assert (bc.variable == w)
assert (bc.order == 0)
assert (bc.normal_component == False)
assert (bc.index_component == [0])
