def test_calls_2d_3():
domain = Square()
V = FunctionSpace('V', domain)
x, y = domain.coordinates
pn = Field('pn', V)
wn = Field('wn', V)
dp = TestFunction(V, name='dp')
dw = TestFunction(V, name='dw')
tau = TestFunction(V, name='tau')
sigma = TestFunction(V, name='sigma')
Re = Constant('Re', real=True)
dt = Constant('dt', real=True)
alpha = Constant('alpha', real=True)
l1 = LinearForm(tau,
bracket(pn, wn) * tau - 1. / Re * dot(grad(tau), grad(wn)))
# ...
l = LinearForm((tau, sigma), dt * l1(tau))
print(evaluate(l, verbose=True))
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_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))
# ...
# ... Poisson's bracket properties
assert bracket(alpha * f, g) == alpha * bracket(f, g)
assert bracket(f, alpha * g) == alpha * bracket(f, g)
assert bracket(f + h, g) == bracket(f, g) + bracket(h, g)
assert bracket(f, g + h) == bracket(f, g) + bracket(f, h)
assert bracket(f, f) == 0
assert bracket(f, g) == -bracket(g, f)
assert bracket(f, g * h) == g * bracket(f, h) + bracket(f, g) * h
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
# Poisson's bracket in 2D
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
u = element_of(V, name='V')
assert bracket(u, 1) == 0 # native int
assert bracket(u, 2.3) == 0 # native float
assert bracket(u, 4 + 5j) == 0 # native complex
assert bracket(u, Integer(1)) == 0 # sympy Integer
assert bracket(u, Float(2.3)) == 0 # sympy Float
assert bracket(u, Rational(6, 7)) == 0 # sympy Rational
assert bracket(u, Constant('a')) == 0 # sympde Constant
assert bracket(1, u) == 0 # native int
assert bracket(2.3, u) == 0 # native float
assert bracket(4 + 5j, u) == 0 # native complex
assert bracket(Integer(1), u) == 0 # sympy Integer
assert bracket(Float(2.3), u) == 0 # sympy Float
assert bracket(Rational(6, 7), u) == 0 # sympy Rational
assert bracket(Constant('a'), u) == 0 # sympde Constant
def test_linear_form_2d_1():
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 = ScalarFunctionSpace('V', domain)
W = VectorFunctionSpace('W', 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)
l = LinearForm(v, int_0(x * y * v))
assert (l.domain == domain.interior)
assert (l(v1) == int_0(x * y * v1))
# ...
# ...
g = Tuple(x**2, y**2)
l = LinearForm(v, int_1(v * dot(g, nn)))
print(l)
assert (l.domain == B1)
assert (l(v1) == int_1(v1 * trace_1(g, B1)))
# ...
# ...
g = Tuple(x**2, y**2)
l = LinearForm(v, int_1(v * dot(g, nn)) + int_0(x * y * v))
assert (len(l.domain.args) == 2)
for i in l.domain.args:
assert (i in [domain.interior, B1])
assert (l(v1) == int_1(v1 * trace_1(g, B1)) + int_0(x * y * v1))
# ...
# ...
l1 = LinearForm(v1, int_0(x * y * v1))
l = LinearForm(v, l1(v))
assert (l.domain == domain.interior)
assert (l(u1) == int_0(x * y * u1))
# ...
# ...
g = Tuple(x, y)
l1 = LinearForm(u1, int_0(x * y * u1))
l2 = LinearForm(u2, int_0(dot(grad(u2), g)))
l = LinearForm(v, l1(v) + l2(v))
assert (l.domain == domain.interior)
assert (l(v1) == int_0(x * y * v1) + int_0(dot(grad(v1), g)))
# ...
# ...
pn, wn = [element_of(V, name=i) for i in ['pn', 'wn']]
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)
l1 = LinearForm(
tau, int_0(bracket(pn, wn) * tau - 1. / Re * dot(grad(tau), grad(wn))))
l = LinearForm((tau, sigma), dt * l1(tau))
assert (l.domain == domain.interior)
assert (l(u1,
u2).expand() == int_0(-1.0 * dt * dot(grad(u1), grad(wn)) / Re) +
int_0(dt * u1 * bracket(pn, wn)))