def test_boundary_3():
Omega_1 = InteriorDomain('Omega_1', dim=2)
Gamma_1 = Boundary(r'\Gamma_1', Omega_1, axis=0, ext=-1)
Gamma_4 = Boundary(r'\Gamma_4', Omega_1, axis=1, ext=1)
Omega = Domain('Omega',
interiors=[Omega_1],
boundaries=[Gamma_1, Gamma_4])
assert(Omega.get_boundary(axis=0, ext=-1) == Gamma_1)
assert(Omega.get_boundary(axis=1, ext=1) == Gamma_4)
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_target_mapping_2d_1():
dim = 2
x1, x2 = symbols('x1, x2')
constants = ['c1', 'c2', 'D', 'k']
c1, c2, D, k = [Constant(i) for i in constants]
M = TargetMapping('M', dim=dim)
domain = M(Domain('Omega', dim=dim))
assert (not (M[0] == x1))
assert (not (M[1] == x2))
assert (LogicalExpr(M[0],
domain) == -D * x1**2 + c1 + x1 * (-k + 1) * cos(x2))
assert (LogicalExpr(M[1], domain) == c2 + x1 * (k + 1) * sin(x2))
assert (LogicalExpr(dx1(M[0]), domain) == -2 * D * x1 + (-k + 1) * cos(x2))
assert (LogicalExpr(dx1(M[1]), domain) == (k + 1) * sin(x2))
assert (LogicalExpr(dx2(M[0]), domain) == -x1 * (-k + 1) * sin(x2))
assert (LogicalExpr(dx2(M[1]), domain) == x1 * (k + 1) * cos(x2))
expected = Matrix(
[[-2 * D * x1 + (-k + 1) * cos(x2), -x1 * (-k + 1) * sin(x2)],
[(k + 1) * sin(x2), x1 * (k + 1) * cos(x2)]])
assert (Jacobian(M) == expected)
def test_tensorize_2d_1_mapping():
DIM = 2
M = Mapping('Map', DIM)
domain = Domain('Omega', dim=DIM)
B1 = Boundary(r'\Gamma_1', domain)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
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), mu*u*v + dot(grad(u),grad(v)))
a = BilinearForm((u, v), int_0(dot(grad(u), grad(v))))
# a = BilinearForm((u,v), dx(u)*v)
# a = BilinearForm((u,v), laplace(u)*laplace(v))
expr = TensorExpr(a, mapping=M)
print(expr)
def test_derivatives_2d_with_mapping():
dim = 2
M = Mapping('M', dim=dim)
O = M(Domain('Omega', dim=dim))
V = ScalarFunctionSpace('V', O, kind='h1')
u = element_of(V, 'u')
det_jac = Jacobian(M).det()
J = Symbol('J')
expr = M
assert SymbolicExpr(expr) == Symbol('M')
expr = M[0]
assert SymbolicExpr(expr) == Symbol('x')
expr = M[1]
assert SymbolicExpr(expr) == Symbol('y')
expr = dx2(M[0])
assert SymbolicExpr(expr) == Symbol('x_x2')
expr = dx1(M[1])
assert SymbolicExpr(expr) == Symbol('y_x1')
expr = LogicalExpr(dx(u), O).subs(det_jac, J)
expected = '(u_x1 * y_x2 - u_x2 * y_x1)/J'
difference = SymbolicExpr(expr) - sympify(expected)
assert difference.simplify() == 0
def test_mapping_2d_2():
print('============ test_mapping_2d_2 ==============')
rdim = 2
F = Mapping('F', rdim)
domain = Domain('Omega', dim=rdim)
D = F(domain)
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_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 = Matrix((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_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_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_latex_2d_3():
DIM = 2
domain = Domain('Omega', dim=DIM)
B1 = Boundary(r'\Gamma_1', domain)
B2 = Boundary(r'\Gamma_2', domain)
B3 = Boundary(r'\Gamma_3', domain)
V = ScalarFunctionSpace('V', domain)
x = V.coordinates
v = element_of(V, name='v')
u = element_of(V, name='u')
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
# ...
expr = dot(grad(v), grad(u))
a_0 = BilinearForm((v, u), int_0(expr))
expr = v * u
a_bnd = BilinearForm((v, u), int_1(expr))
expr = a_0(v, u) + a_bnd(v, u)
a = BilinearForm((v, u), expr)
print(latex(a_0))
print(latex(a_bnd))
print(latex(a))
# print(a)
print('')
def test_terminal_expr_bilinear_3d_1():
domain = Domain('Omega', dim=3)
M = Mapping('M', 3)
mapped_domain = M(domain)
V = ScalarFunctionSpace('V', domain)
VM = ScalarFunctionSpace('VM', mapped_domain)
u, v = elements_of(V, names='u,v')
um, vm = elements_of(VM, names='u,v')
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(mapped_domain, expr)
J = M.det_jacobian
det = dx1(M[0])*dx2(M[1])*dx3(M[2]) - dx1(M[0])*dx2(M[2])*dx3(M[1]) - dx1(M[1])*dx2(M[0])*dx3(M[2])\
+ dx1(M[1])*dx2(M[2])*dx3(M[0]) + dx1(M[2])*dx2(M[0])*dx3(M[1]) - dx1(M[2])*dx2(M[1])*dx3(M[0])
a1 = BilinearForm((u, v), int_0(dot(grad(u), grad(v))))
a2 = BilinearForm((um, vm), int_1(dot(grad(um), grad(vm))))
a3 = BilinearForm((u, v), int_0(J * dot(grad(u), grad(v))))
e1 = TerminalExpr(a1)
e2 = TerminalExpr(a2)
e3 = TerminalExpr(a3)
assert e1[0].expr == dx1(u) * dx1(v) + dx2(u) * dx2(v) + dx3(u) * dx3(v)
assert e2[0].expr == dx(um) * dx(vm) + dy(um) * dy(vm) + dz(um) * dz(vm)
assert e3[0].expr.factor() == (dx1(u) * dx1(v) + dx2(u) * dx2(v) +
dx3(u) * dx3(v)) * det
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_partial_derivatives_1():
print('============ test_partial_derivatives_1 ==============')
# ...
domain = Domain('Omega', dim=2)
x,y = domain.coordinates
V = ScalarFunctionSpace('V', domain)
F,u,v,w = [ScalarField(V, name=i) for i in ['F', 'u', 'v', 'w']]
uvw = Tuple(u,v,w)
alpha = Constant('alpha')
beta = Constant('beta')
# ...
# ...
assert(dx(x**2) == 2*x)
assert(dy(x**2) == 0)
assert(dz(x**2) == 0)
assert(dx(x*F) == F + x*dx(F))
assert(dx(uvw) == Matrix([[dx(u), dx(v), dx(w)]]))
assert(dx(uvw) + dy(uvw) == Matrix([[dx(u) + dy(u),
dx(v) + dy(v),
dx(w) + dy(w)]]))
expected = Matrix([[alpha*dx(u) + beta*dy(u),
alpha*dx(v) + beta*dy(v),
alpha*dx(w) + beta*dy(w)]])
assert(alpha * dx(uvw) + beta * dy(uvw) == expected)
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_calculus_2d_4():
DIM = 2
domain = Domain('Omega', dim=DIM)
V = ScalarFunctionSpace('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))
# ...
# ... plus operator
assert (plus(u + v) == plus(u) + plus(v))
assert (plus(a * u) == a * plus(u))
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
def test_derivatives_2d_without_mapping():
O = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', O)
u = element_of(V, 'u')
expr = u
assert SymbolicExpr(expr) == Symbol('u')
expr = dx(u)
assert SymbolicExpr(expr) == Symbol('u_x')
expr = dx(dx(u))
assert SymbolicExpr(expr) == Symbol('u_xx')
expr = dx(dy(u))
assert SymbolicExpr(expr) == Symbol('u_xy')
expr = dy(dx(u))
assert SymbolicExpr(expr) == Symbol('u_xy')
expr = dy(dx(dz(u)))
assert SymbolicExpr(expr) == Symbol('u_xyz')
expr = dy(dy(dx(u)))
assert SymbolicExpr(expr) == Symbol('u_xyy')
expr = dy(dz(dy(u)))
assert SymbolicExpr(expr) == Symbol('u_yyz')
def test_linear_expr_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)
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']]
# ...
l = LinearExpr(v, x * y * v)
print(l)
print(l.expr)
print(l(v1))
# TODO
# print(l(v1+v2))
print('')
# ...
# ...
l = LinearExpr((v1, v2), x * v1 + y * v2)
print(l)
print(l.expr)
print(l(u1, u2))
# TODO
# print(l(u1+v1, u2+v2))
print('')
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_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_generic_line():
# Create 1D domain (Line) from interval [-3, 4]
domain = Line('line', bounds=(-3, 4))
assert isinstance(domain, Line)
# BasicDomain's attributes
assert domain.dim == 1
assert domain.name == 'line'
assert domain.coordinates == x1
# Domain's attributes
assert isinstance(domain.interior, NCubeInterior)
assert len(domain.boundary) == 2
assert domain.dtype == {'type': 'Line', 'parameters': {'bounds': [-3, 4]}}
# NCube's attributes
assert domain.min_coords == (-3, )
assert domain.max_coords == (4, )
# Line's attributes
assert domain.bounds == (-3, 4)
# Export to file, read it again and compare with original domain
domain.export('domain.h5')
D = Domain.from_file('domain.h5')
assert D.logical_domain == domain
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_polar_mapping_2d_1():
dim = 2
x1, x2 = symbols('x1, x2')
constants = ['c1', 'c2', 'rmax', 'rmin']
c1, c2, rmax, rmin = [Constant(i) for i in constants]
M = PolarMapping('M', dim=dim)
domain = M(Domain('Omega', dim=dim))
assert (not (M[0] == x1))
assert (not (M[1] == x2))
assert (LogicalExpr(M[0], domain) == c1 + (rmax * x1 + rmin *
(-x1 + 1)) * cos(x2))
assert (LogicalExpr(M[1], domain) == c2 + (rmax * x1 + rmin *
(-x1 + 1)) * sin(x2))
assert (LogicalExpr(dx1(M[0]), domain) == (rmax - rmin) * cos(x2))
assert (LogicalExpr(dx1(M[1]), domain) == (rmax - rmin) * sin(x2))
expected = -(rmax * x1 + rmin * (-x1 + 1)) * sin(x2)
assert (expand(LogicalExpr(dx2(M[0]), domain)) == expand(expected))
assert (LogicalExpr(dx2(M[1]),
domain) == (rmax * x1 + rmin * (-x1 + 1)) * cos(x2))
expected = Matrix([[(rmax - rmin) * cos(x2),
-(rmax * x1 + rmin * (-x1 + 1)) * sin(x2)],
[(rmax - rmin) * sin(x2),
(rmax * x1 + rmin * (-x1 + 1)) * cos(x2)]])
assert (expand(LogicalExpr(Jacobian(M), domain)) == expand(expected))
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_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_boundary_2d_2():
Omega_1 = InteriorDomain('Omega_1', dim=2)
B1 = Boundary('B1', Omega_1)
B2 = Boundary('B2', Omega_1)
B3 = Boundary('B3', Omega_1)
domain = Domain('Omega', interiors=[Omega_1], boundaries=[B1, B2, B3])
V = FunctionSpace('V', domain)
v = TestFunction(V, name='v')
u = TestFunction(V, name='u')
x, y = V.coordinates
alpha = Constant('alpha')
# ...
print('==== l0 ====')
l0 = LinearForm(v, x * y * v, name='l0')
print(evaluate(l0, verbose=VERBOSE))
print('')
# ...
# ...
print('==== l1 ====')
g = Tuple(x**2, y**2)
l1 = LinearForm(v, v * trace_1(g, domain.boundary))
print(evaluate(l1, verbose=VERBOSE))
print('')
# ...
# ...
print('==== l2 ====')
B_neumann = Union(B1, B2)
g = Tuple(x**2, y**2)
l2 = LinearForm(v, v * trace_1(g, B_neumann), name='l2')
print(evaluate(l2, verbose=VERBOSE))
print('')
# ...
# ...
print('==== l3 ====')
l3 = LinearForm(v, l2(v))
assert (l3(v).__str__ == l2(v).__str__)
print(evaluate(l3, verbose=VERBOSE))
print('')
# ...
# ...
print('==== l4 ====')
l4 = LinearForm(v, l0(v) + l2(v))
print(evaluate(l4, verbose=VERBOSE))
print('')
