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_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_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_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_linearize_form_2d_2():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
v, F, u = elements_of(V, names='v, F, u')
int_0 = lambda expr: integral(domain, expr)
# ...
l1 = LinearForm(v, int_0(F**2 * v))
l = LinearForm(v, l1(v))
a = linearize(l, F, trials=u)
expected = linearize(l1, F, trials=u)
assert a == expected
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_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_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_linearize_form_2d_1():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
W = VectorFunctionSpace('W', domain)
v, F, u = elements_of(V, names='v, F, u')
w, G, m = elements_of(W, names='w, G, m')
int_0 = lambda expr: integral(domain, expr)
# ...
l = LinearForm(v, int_0(F**2 * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(2 * F * u * v)
# ...
# ...
l = LinearForm(v, int_0(dot(grad(F), grad(F)) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(2 * dot(grad(F), grad(u)) * v)
# ...
# ...
l = LinearForm(v, int_0(exp(-F) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(-exp(-F) * u * v)
# ...
# ...
l = LinearForm(v, int_0(cos(F) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(-sin(F) * u * v)
# ...
# ...
l = LinearForm(v, int_0(cos(F**2) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(-2 * F * sin(F**2) * u * v)
# ...
# ...
l = LinearForm(v, int_0(F**2 * dot(grad(F), grad(v))))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(2 * F * u * dot(grad(F), grad(v)) +
F**2 * dot(grad(u), grad(v)))
# ...
# ...
l = LinearForm(w, int_0(dot(rot(G), grad(G)) * w))
a = linearize(l, G, trials=m)
assert a(m, w) == int_0((dot(rot(m), grad(G)) + dot(rot(G), grad(m))) * w)
def test_terminal_expr_linear_2d_4():
D1 = InteriorDomain('D1', dim=2)
D2 = InteriorDomain('D2', dim=2)
domain = Union(D1, D2)
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']]
# ...
int_0 = lambda expr: integral(domain, expr)
l = LinearForm(v, int_0(x * y * v))
print(TerminalExpr(l))
print('')
def test_terminal_expr_linear_2d_3():
domain = Square()
B = domain.boundary
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
nn = NormalVector('nn')
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']]
# ...
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B, expr)
l = LinearForm(v, int_1(dot(grad(v), nn)))
print(TerminalExpr(l))
print('')
def test_linear_form_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)))
assert (l.domain == domain.interior)
assert (l(v1) == int_0(dot(g, v1)))
# ...
# ...
g = Matrix((x, y))
l1 = LinearForm(v1, int_0(dot(g, v1)))
l = LinearForm(v, l1(v))
assert (l.domain == domain.interior)
assert (l(u1) == int_0(dot(g, u1)))
# ...
# ...
g1 = Matrix((x, 0))
g2 = Matrix((0, y))
l1 = LinearForm(v1, int_0(dot(v1, g1)))
l2 = LinearForm(v2, int_0(dot(v2, g2)))
l = LinearForm(v, l1(v) + l2(v))
assert (l.domain == domain.interior)
assert (l(u) == int_0(dot(u, g1)) + int_0(dot(u, g2)))
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)
eps = Constant('eps', real=True)
V = FunctionSpace('V', domain)
u, u1, u2 = [TestFunction(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [TestFunction(V, name=i) for i in ['v', 'v1', 'v2']]
g = Tuple(x**2, y**2)
# ...
d_forms = {}
d_forms['l1'] = LinearForm(v, x * y * v)
d_forms['l2'] = LinearForm(v, v * trace_1(g, B1))
d_forms['l3'] = LinearForm(v, v * trace_1(g, B1) + x * y * v)
# nitsche
f = Function('f')
g = Function('g')
l0 = LinearForm(v, f(x, y) * v)
l_B1 = LinearForm(
v,
g(x, y) * trace_0(v, B1) / eps -
kappa * g(x, y) * trace_1(grad(v), B1))
d_forms['l4'] = LinearForm(v, l0(v) + l_B1(v))
# ...
# ...
for name, expr in d_forms.items():
export(expr, 'linform_2d_{}.png'.format(name))
def eval(cls, expr, domain, **options):
"""."""
from sympde.expr.evaluation import TerminalExpr, DomainExpression
from sympde.expr.expr import BilinearForm, LinearForm, BasicForm, Norm
from sympde.expr.expr import Integral
types = (ScalarFunction, VectorFunction, DifferentialOperator, Trace,
Integral)
mapping = domain.mapping
dim = domain.dim
assert mapping
# TODO this is not the dim of the domain
l_coords = ['x1', 'x2', 'x3'][:dim]
ph_coords = ['x', 'y', 'z']
if not has(expr, types):
if has(expr, DiffOperator):
return cls(expr, domain, evaluate=False)
else:
syms = symbols(ph_coords[:dim])
if isinstance(mapping, InterfaceMapping):
mapping = mapping.minus
# here we assume that the two mapped domains
# are identical in the interface so we choose one of them
Ms = [mapping[i] for i in range(dim)]
expr = expr.subs(list(zip(syms, Ms)))
if mapping.is_analytical:
expr = expr.subs(list(zip(Ms, mapping.expressions)))
return expr
if isinstance(expr, Symbol) and expr.name in l_coords:
return expr
if isinstance(expr, Symbol) and expr.name in ph_coords:
return mapping[ph_coords.index(expr.name)]
elif isinstance(expr, Add):
args = [cls.eval(a, domain) for a in expr.args]
v = S.Zero
for i in args:
v += i
n, d = v.as_numer_denom()
return n / d
elif isinstance(expr, Mul):
args = [cls.eval(a, domain) for a in expr.args]
v = S.One
for i in args:
v *= i
return v
elif isinstance(expr, _logical_partial_derivatives):
if mapping.is_analytical:
Ms = [mapping[i] for i in range(dim)]
expr = expr.subs(list(zip(Ms, mapping.expressions)))
return expr
elif isinstance(expr, IndexedVectorFunction):
el = cls.eval(expr.base, domain)
el = TerminalExpr(el, domain=domain.logical_domain)
return el[expr.indices[0]]
elif isinstance(expr, MinusInterfaceOperator):
mapping = mapping.minus
newexpr = PullBack(expr.args[0], mapping)
test = newexpr.test
newexpr = newexpr.expr.subs(test, MinusInterfaceOperator(test))
return newexpr
elif isinstance(expr, PlusInterfaceOperator):
mapping = mapping.plus
newexpr = PullBack(expr.args[0], mapping)
test = newexpr.test
newexpr = newexpr.expr.subs(test, PlusInterfaceOperator(test))
return newexpr
elif isinstance(expr, (VectorFunction, ScalarFunction)):
return PullBack(expr, mapping).expr
elif isinstance(expr, Transpose):
arg = cls(expr.arg, domain)
return Transpose(arg)
elif isinstance(expr, grad):
arg = expr.args[0]
if isinstance(mapping, InterfaceMapping):
if isinstance(arg, MinusInterfaceOperator):
a = arg.args[0]
mapping = mapping.minus
elif isinstance(arg, PlusInterfaceOperator):
a = arg.args[0]
mapping = mapping.plus
else:
raise TypeError(arg)
arg = type(arg)(cls.eval(a, domain))
else:
arg = cls.eval(arg, domain)
return mapping.jacobian.inv().T * grad(arg)
elif isinstance(expr, curl):
arg = expr.args[0]
if isinstance(mapping, InterfaceMapping):
if isinstance(arg, MinusInterfaceOperator):
arg = arg.args[0]
mapping = mapping.minus
elif isinstance(arg, PlusInterfaceOperator):
arg = arg.args[0]
mapping = mapping.plus
else:
raise TypeError(arg)
if isinstance(arg, VectorFunction):
arg = PullBack(arg, mapping)
else:
arg = cls.eval(arg, domain)
if isinstance(arg, PullBack) and isinstance(
arg.kind, HcurlSpaceType):
J = mapping.jacobian
arg = arg.test
if isinstance(expr.args[0],
(MinusInterfaceOperator, PlusInterfaceOperator)):
arg = type(expr.args[0])(arg)
if expr.is_scalar:
return (1 / J.det()) * curl(arg)
return (J / J.det()) * curl(arg)
else:
raise NotImplementedError('TODO')
elif isinstance(expr, div):
arg = expr.args[0]
if isinstance(mapping, InterfaceMapping):
if isinstance(arg, MinusInterfaceOperator):
arg = arg.args[0]
mapping = mapping.minus
elif isinstance(arg, PlusInterfaceOperator):
arg = arg.args[0]
mapping = mapping.plus
else:
raise TypeError(arg)
if isinstance(arg, (ScalarFunction, VectorFunction)):
arg = PullBack(arg, mapping)
else:
arg = cls.eval(arg, domain)
if isinstance(arg, PullBack) and isinstance(
arg.kind, HdivSpaceType):
J = mapping.jacobian
arg = arg.test
if isinstance(expr.args[0],
(MinusInterfaceOperator, PlusInterfaceOperator)):
arg = type(expr.args[0])(arg)
return (1 / J.det()) * div(arg)
elif isinstance(arg, PullBack):
return SymbolicTrace(mapping.jacobian.inv().T * grad(arg.test))
else:
raise NotImplementedError('TODO')
elif isinstance(expr, laplace):
arg = expr.args[0]
v = cls.eval(grad(arg), domain)
v = mapping.jacobian.inv().T * grad(v)
return SymbolicTrace(v)
# elif isinstance(expr, hessian):
# arg = expr.args[0]
# if isinstance(mapping, InterfaceMapping):
# if isinstance(arg, MinusInterfaceOperator):
# arg = arg.args[0]
# mapping = mapping.minus
# elif isinstance(arg, PlusInterfaceOperator):
# arg = arg.args[0]
# mapping = mapping.plus
# else:
# raise TypeError(arg)
# v = cls.eval(grad(expr.args[0]), domain)
# v = mapping.jacobian.inv().T*grad(v)
# return v
elif isinstance(expr, (dot, inner, outer)):
args = [cls.eval(arg, domain) for arg in expr.args]
return type(expr)(*args)
elif isinstance(expr, _diff_ops):
raise NotImplementedError('TODO')
# TODO MUST BE MOVED AFTER TREATING THE CASES OF GRAD, CURL, DIV IN FEEC
elif isinstance(expr, (Matrix, ImmutableDenseMatrix)):
n_rows, n_cols = expr.shape
lines = []
for i_row in range(0, n_rows):
line = []
for i_col in range(0, n_cols):
line.append(cls.eval(expr[i_row, i_col], domain))
lines.append(line)
return type(expr)(lines)
elif isinstance(expr, dx):
if expr.atoms(PlusInterfaceOperator):
mapping = mapping.plus
elif expr.atoms(MinusInterfaceOperator):
mapping = mapping.minus
arg = expr.args[0]
arg = cls(arg, domain, evaluate=True)
if isinstance(arg, PullBack):
arg = TerminalExpr(arg, domain=domain.logical_domain)
elif isinstance(arg, MatrixElement):
arg = TerminalExpr(arg, domain=domain.logical_domain)
# ...
if dim == 1:
lgrad_arg = LogicalGrad_1d(arg)
if not isinstance(lgrad_arg, (list, tuple, Tuple, Matrix)):
lgrad_arg = Tuple(lgrad_arg)
elif dim == 2:
lgrad_arg = LogicalGrad_2d(arg)
elif dim == 3:
lgrad_arg = LogicalGrad_3d(arg)
grad_arg = Covariant(mapping, lgrad_arg)
expr = grad_arg[0]
return expr
elif isinstance(expr, dy):
if expr.atoms(PlusInterfaceOperator):
mapping = mapping.plus
elif expr.atoms(MinusInterfaceOperator):
mapping = mapping.minus
arg = expr.args[0]
arg = cls(arg, domain, evaluate=True)
if isinstance(arg, PullBack):
arg = TerminalExpr(arg, domain=domain.logical_domain)
elif isinstance(arg, MatrixElement):
arg = TerminalExpr(arg, domain=domain.logical_domain)
# ..p
if dim == 1:
lgrad_arg = LogicalGrad_1d(arg)
elif dim == 2:
lgrad_arg = LogicalGrad_2d(arg)
elif dim == 3:
lgrad_arg = LogicalGrad_3d(arg)
grad_arg = Covariant(mapping, lgrad_arg)
expr = grad_arg[1]
return expr
elif isinstance(expr, dz):
if expr.atoms(PlusInterfaceOperator):
mapping = mapping.plus
elif expr.atoms(MinusInterfaceOperator):
mapping = mapping.minus
arg = expr.args[0]
arg = cls(arg, domain, evaluate=True)
if isinstance(arg, PullBack):
arg = TerminalExpr(arg, domain=domain.logical_domain)
elif isinstance(arg, MatrixElement):
arg = TerminalExpr(arg, domain=domain.logical_domain)
# ...
if dim == 1:
lgrad_arg = LogicalGrad_1d(arg)
elif dim == 2:
lgrad_arg = LogicalGrad_2d(arg)
elif dim == 3:
lgrad_arg = LogicalGrad_3d(arg)
grad_arg = Covariant(mapping, lgrad_arg)
expr = grad_arg[2]
return expr
elif isinstance(expr, (Symbol, Indexed)):
return expr
elif isinstance(expr, NormalVector):
return expr
elif isinstance(expr, Pow):
b = expr.base
e = expr.exp
expr = Pow(cls(b, domain), cls(e, domain))
return expr
elif isinstance(expr, Trace):
e = cls.eval(expr.expr, domain)
bd = expr.boundary.logical_domain
order = expr.order
return Trace(e, bd, order)
elif isinstance(expr, Integral):
domain = expr.domain
mapping = domain.mapping
assert domain is not None
if expr.is_domain_integral:
J = mapping.jacobian
det = sqrt((J.T * J).det())
else:
axis = domain.axis
J = JacobianSymbol(mapping, axis=axis)
det = sqrt((J.T * J).det())
body = cls.eval(expr.expr, domain) * det
domain = domain.logical_domain
return Integral(body, domain)
elif isinstance(expr, BilinearForm):
tests = [get_logical_test_function(a) for a in expr.test_functions]
trials = [
get_logical_test_function(a) for a in expr.trial_functions
]
body = cls.eval(expr.expr, domain)
return BilinearForm((trials, tests), body)
elif isinstance(expr, LinearForm):
tests = [get_logical_test_function(a) for a in expr.test_functions]
body = cls.eval(expr.expr, domain)
return LinearForm(tests, body)
elif isinstance(expr, Norm):
kind = expr.kind
exponent = expr.exponent
e = cls.eval(expr.expr, domain)
domain = domain.logical_domain
norm = Norm(e, domain, kind, evaluate=False)
norm._exponent = exponent
return norm
elif isinstance(expr, DomainExpression):
domain = expr.target
J = domain.mapping.jacobian
newexpr = cls.eval(expr.expr, domain)
newexpr = TerminalExpr(newexpr, domain=domain)
domain = domain.logical_domain
det = TerminalExpr(sqrt((J.T * J).det()), domain=domain)
return DomainExpression(domain,
ImmutableDenseMatrix([[newexpr * det]]))
elif isinstance(expr, Function):
args = [cls.eval(a, domain) for a in expr.args]
return type(expr)(*args)
return cls(expr, domain, evaluate=False)
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)))
def test_linearity_linear_form_2d_1():
from sympde.expr.errors import UnconsistentLinearExpressionError
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)
v, v1, v2 = elements_of(V, names='v, v1, v2')
w = element_of(W, name='w')
# ...
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
# The following integral expressions are linear, hence it must be possible
# to create LinearForm objects from them
_ = LinearForm(v, int_0(x * y * v))
_ = LinearForm(v, int_0(x * y * v + v))
g = Matrix((x**2, y**2))
_ = LinearForm(v, int_0(v * dot(g, nn)))
g = Matrix((x**2, y**2))
_ = LinearForm(v, int_1(v * dot(g, nn)) + int_0(x * y * v))
l1 = LinearForm(v1, int_0(x * y * v1))
_ = LinearForm(v, l1(v))
g = Matrix((x, y))
l1 = LinearForm(v1, int_0(x * y * v1))
l2 = LinearForm(v2, int_0(dot(grad(v2), g)))
_ = LinearForm(v, l1(v) + l2(v))
l1 = LinearForm(v1, int_0(x * y * v1))
l2 = LinearForm(v1, int_0(v1))
_ = LinearForm(v, l1(v) + kappa * l2(v))
g = Matrix((x**2, y**2))
l1 = LinearForm(v1, int_0(x * y * v1))
l2 = LinearForm(v1, int_0(v1))
l3 = LinearForm(v, int_1(v * dot(g, nn)))
_ = LinearForm(v, l1(v) + kappa * l2(v) + mu * l3(v))
l1 = LinearForm(w, int_0(5 * w * x))
l2 = LinearForm(w, int_1(w * y))
_ = LinearForm(w, l1(w) + kappa * l2(w))
l0 = LinearForm(v, int_0(x * y * v))
l1 = LinearForm(w, int_0(w[0] * y))
l2 = LinearForm(w, int_1(w[1] * x))
l3 = LinearForm(w, kappa * l1(w) + mu * l2(w))
_ = LinearForm((v, w), l0(v) + l3(w))
# The following integral expressions are not linear, hence LinearForm must
# raise an exception
with pytest.raises(UnconsistentLinearExpressionError):
_ = LinearForm(v, int_0(x * y * v + 1))
with pytest.raises(UnconsistentLinearExpressionError):
_ = LinearForm(v, int_0(x * v**2))
with pytest.raises(UnconsistentLinearExpressionError):
_ = LinearForm(v, int_0(x * y * v) + int_1(v * exp(v)))
with pytest.raises(UnconsistentLinearExpressionError):
_ = LinearForm(w, int_0(w * w))
with pytest.raises(UnconsistentLinearExpressionError):
_ = LinearForm(w, int_0(w[0] * w[1]))
with pytest.raises(UnconsistentLinearExpressionError):
_ = LinearForm((v, w), int_0(x * w[0]) + int_1(v * w[1]))
def test_stabilization_2d_1():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
b1 = 1.
b2 = 0.
b = Matrix((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
l = 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)
# ...
print(a1)
print(TerminalExpr(a1))
print('')
print(a2)
print(TerminalExpr(a2))
print('')
print(a3)
print(TerminalExpr(a3))
print('')
def test_terminal_expr_linear_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)
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))
print(TerminalExpr(l))
print('')
# ...
# ...
l = LinearForm(v, int_0(x * y * v + v))
print(TerminalExpr(l))
print('')
# ...
# ...
g = Matrix((x**2, y**2))
l = LinearForm(v, int_1(v * dot(g, nn)))
print(TerminalExpr(l))
print('')
# ...
# ...
g = Matrix((x**2, y**2))
l = LinearForm(v, int_1(v * dot(g, nn)) + int_0(x * y * v))
print(TerminalExpr(l))
print('')
# ...
# ...
l1 = LinearForm(v1, int_0(x * y * v1))
l = LinearForm(v, l1(v))
print(TerminalExpr(l))
print('')
# ...
# ...
g = Matrix((x, y))
l1 = LinearForm(v1, int_0(x * y * v1))
l2 = LinearForm(v2, int_0(dot(grad(v2), g)))
l = LinearForm(v, l1(v) + l2(v))
print(TerminalExpr(l))
print('')
# ...
# ...
l1 = LinearForm(v1, int_0(x * y * v1))
l2 = LinearForm(v1, int_0(v1))
l = LinearForm(v, l1(v) + kappa * l2(v))
print(TerminalExpr(l))
print('')
# ...
# ...
g = Matrix((x**2, y**2))
l1 = LinearForm(v1, int_0(x * y * v1))
l2 = LinearForm(v1, int_0(v1))
l3 = LinearForm(v, int_1(v * dot(g, nn)))
l = LinearForm(v, l1(v) + kappa * l2(v) + mu * l3(v))
print(TerminalExpr(l))
print('')
