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_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_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_equation_2d_2():
domain = Square()
V = ScalarFunctionSpace('V', domain)
x, y = domain.coordinates
pn, wn = [element_of(V, name=i) for i in ['pn', 'wn']]
dp = element_of(V, name='dp')
dw = element_of(V, name='dw')
tau = element_of(V, name='tau')
sigma = element_of(V, name='sigma')
Re = Constant('Re', real=True)
dt = Constant('dt', real=True)
alpha = Constant('alpha', real=True)
int_0 = lambda expr: integral(domain, expr)
s = BilinearForm((tau, sigma), int_0(dot(grad(tau), grad(sigma))))
m = BilinearForm((tau, sigma), int_0(tau * sigma))
b1 = BilinearForm((tau, dw), int_0(bracket(pn, dw) * tau))
b2 = BilinearForm((tau, dp), int_0(bracket(dp, wn) * tau))
l1 = LinearForm(
tau, int_0(bracket(pn, wn) * tau - 1. / Re * dot(grad(tau), grad(wn))))
expr = m(tau, dw) - alpha * dt * b1(tau, dw) - dt * b2(
tau, dp) - (alpha * dt / Re) * s(tau, dw)
a = BilinearForm(((tau, sigma), (dp, dw)), int_0(expr))
l = LinearForm((tau, sigma), dt * l1(tau))
bc = [EssentialBC(dp, 0, domain.boundary)]
bc += [EssentialBC(dw, 0, domain.boundary)]
equation = Equation(a, l, tests=[tau, sigma], trials=[dp, dw], bc=bc)
def test_equation_2d_6():
domain = Square()
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 = ScalarFunctionSpace('V', domain)
u, v = [element_of(V, name=i) for i in ['u', 'v']]
int_0 = lambda expr: integral(domain, expr)
# ...
expr = kappa * dot(grad(u), grad(v)) + dot(b, grad(u)) * v
a = BilinearForm((v, u), int_0(expr))
# ...
# ...
expr = f * v
l0 = LinearForm(v, int_0(expr))
# ...
# ...
expr = (-kappa * laplace(u) + dot(b, grad(u))) * dot(b, grad(v))
s1 = BilinearForm((v, u), int_0(expr))
expr = -f * dot(b, grad(v))
l1 = LinearForm(v, int_0(expr))
# ...
# ...
expr = (-kappa * laplace(u) + dot(b, grad(u))) * (dot(b, grad(v)) -
kappa * laplace(v))
s2 = BilinearForm((v, u), int_0(expr))
expr = -f * (dot(b, grad(v)) - kappa * laplace(v))
l2 = LinearForm(v, int_0(expr))
# ...
# ...
expr = (-kappa * laplace(u) + dot(b, grad(u))) * (dot(b, grad(v)) +
kappa * laplace(v))
s3 = BilinearForm((v, u), int_0(expr))
expr = -f * (dot(b, grad(v)) + kappa * laplace(v))
l3 = LinearForm(v, int_0(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)
# ...
bc = EssentialBC(u, 0, domain.boundary)
# ...
l = LinearForm(v, l0(v) + mu * area * l1(v))
eq_1 = Equation(a1, l, tests=v, trials=u, bc=bc)
# ...
# ...
l = LinearForm(v, l0(v) + mu * area * l2(v))
eq_2 = Equation(a2, l, tests=v, trials=u, bc=bc)
# ...
# ...
l = LinearForm(v, l0(v) + mu * area * l3(v))
eq_3 = Equation(a3, l, tests=v, trials=u, bc=bc)
def test_equation_2d_1():
V = ScalarFunctionSpace('V', domain)
U = ScalarFunctionSpace('U', domain)
v = element_of(V, name='v')
u = element_of(U, name='u')
x, y = domain.coordinates
alpha = Constant('alpha')
kappa = Constant('kappa', real=True)
eps = Constant('eps', real=True)
nn = NormalVector('nn')
B1 = Boundary(r'\Gamma_1', domain)
B2 = Boundary(r'\Gamma_2', domain)
B3 = Boundary(r'\Gamma_3', domain)
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
int_2 = lambda expr: integral(B2, expr)
# ... bilinear/linear forms
expr = dot(grad(v), grad(u))
a1 = BilinearForm((v, u), int_0(expr))
expr = v * u
a2 = BilinearForm((v, u), int_0(expr))
expr = v * u
a_B1 = BilinearForm((v, u), int_1(expr))
expr = x * y * v
l1 = LinearForm(v, int_0(expr))
expr = cos(x + y) * v
l2 = LinearForm(v, int_0(expr))
expr = x * y * v
l_B2 = LinearForm(v, int_2(expr))
# ...
# # ...
# with pytest.raises(UnconsistentLhsError):
# equation = Equation(a1, l1(v))
#
# with pytest.raises(UnconsistentLhsError):
# equation = Equation(l1(v), l1(v))
#
# with pytest.raises(UnconsistentLhsError):
# equation = Equation(a1(v,u) + alpha*a2(v,u), l1(v))
# # ...
#
# # ...
# with pytest.raises(UnconsistentRhsError):
# equation = Equation(a1(v,u), l1)
#
# with pytest.raises(UnconsistentRhsError):
# equation = Equation(a1(v,u), a1(v,u))
#
# with pytest.raises(UnconsistentRhsError):
# equation = Equation(a1(v,u), l1(v) + l2(v))
# # ...
# ...
equation = Equation(a1, l1, tests=v, trials=u)
# ...
# ...
a = BilinearForm((v, u), a1(v, u) + alpha * a2(v, u))
equation = Equation(a, l1, tests=v, trials=u)
# ...
# ...
a = BilinearForm((v, u), a1(v, u) + a_B1(v, u))
equation = Equation(a, l1, tests=v, trials=u)
# ...
# ...
a = BilinearForm((v, u), a1(v, u) + a_B1(v, u))
l = LinearForm(v, l1(v) + l2(v))
equation = Equation(a, l, tests=v, trials=u)
# ...
# ...
a = BilinearForm((v, u), a1(v, u) + a_B1(v, u))
l = LinearForm(v, l1(v) + alpha * l_B2(v))
equation = Equation(a, l, tests=v, trials=u)
# ...
# ... using bc
equation = Equation(a1, l1, tests=v, trials=u, bc=EssentialBC(u, 0, B1))
# ...
# ... Poisson with Nitsch method
g = cos(pi * x) * cos(pi * y)
a0 = BilinearForm((u, v), int_0(dot(grad(u), grad(v))))
a_B1 = BilinearForm((u, v), -int_1(kappa * u * dot(grad(v), nn) -
v * dot(grad(u), nn) + u * v / eps))
a = BilinearForm((u, v), a0(u, v) + a_B1(u, v))
l = LinearForm(v, int_0(g * v / eps) - int_1(kappa * v * g))
equation = Equation(a, l, tests=v, trials=u)