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_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_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_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_assembly_linear_form_2d_1():
b = LinearForm(v, integral(domain, v*cos(x)))
ast = AST(b)
stmt = parse(ast.expr, settings={'dim': ast.dim, 'nderiv': ast.nderiv})
print(pycode(stmt))
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_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_logical_expr_2d_3():
dim = 2
A = Square('A')
B = Square('B')
M1 = Mapping('M1', dim=dim)
M2 = Mapping('M2', dim=dim)
D1 = M1(A)
D2 = M2(B)
domain = D1.join(D2,
name='domain',
bnd_minus=D1.get_boundary(axis=0, ext=1),
bnd_plus=D2.get_boundary(axis=0, ext=-1))
V = VectorFunctionSpace('V', domain, kind='hcurl')
u, v = [element_of(V, name=i) for i in ['u', 'v']]
int_0 = lambda expr: integral(domain, expr)
expr = LogicalExpr(int_0(dot(u, v)), domain)
assert str(
expr.args[0]
) == 'Integral(A, Dot((Jacobian(M1)**(-1)).T * u, (Jacobian(M1)**(-1)).T * v)*sqrt(det(Jacobian(M1).T * Jacobian(M1))))'
assert str(
expr.args[1]
) == 'Integral(B, Dot((Jacobian(M2)**(-1)).T * u, (Jacobian(M2)**(-1)).T * v)*sqrt(det(Jacobian(M2).T * Jacobian(M2))))'
def test_newton_2d_1():
# ... abstract model
B1 = Boundary(r'\Gamma_1', domain)
V = ScalarFunctionSpace('V', domain)
x, y = domain.coordinates
Un = element_of(V, name='Un')
v = element_of(V, name='v')
int_0 = lambda expr: integral(domain, expr)
f = -4. * exp(-Un)
l = LinearForm(v, int_0(dot(grad(v), grad(Un)) - f * v))
u = element_of(V, name='u')
eq = NewtonIteration(l, Un, trials=u)
# ...
# ...
expected = int_0(-4.0 * u * v * exp(-Un) + dot(grad(u), grad(v)))
assert (eq.lhs.expr == expected)
# ...
# ...
bc = EssentialBC(u, 0, B1)
eq = NewtonIteration(l, Un, bc=bc, trials=u)
def test_assembly_bilinear_form_2d_1():
a = BilinearForm((u,v), integral(domain, dot(grad(u), grad(v))))
ast = AST(a)
stmt = parse(ast.expr, settings={'dim': ast.dim, 'nderiv': ast.nderiv})
print(pycode(stmt))
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_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_latex_2d_4():
DIM = 2
domain = Domain('Omega', dim=DIM)
# ... abstract model
V = VectorFunctionSpace('V', domain)
W = ScalarFunctionSpace('W', domain)
v = element_of(V, name='v')
u = element_of(V, name='u')
p = element_of(W, name='p')
q = element_of(W, name='q')
int_0 = lambda expr: integral(domain, expr)
a = BilinearForm((v, u), int_0(inner(grad(v), grad(u))))
b = BilinearForm((v, p), int_0(div(v) * p))
A = BilinearForm(((v, q), (u, p)), a(v, u) - b(v, p) + b(u, q))
# ...
print(latex(A))
# print(latex(tensorize(A)))
print('')
def test_latex_2d_1():
DIM = 2
domain = Domain('Omega', dim=DIM)
V = ScalarFunctionSpace('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(grad(v)) == r'\nabla{v}')
assert (latex(dot(grad(v), grad(u))) == r'\nabla{u} \cdot \nabla{v}')
a = BilinearForm((v, u), int_0(dot(grad(v), grad(u))))
print(latex(a))
# assert(latex(a) == r'\int_{0}^{1}\int_{0}^{1} \nabla{v} \cdot \nabla{u} dxdy')
b = LinearForm(v, int_0(sin(pi * x) * cos(pi * y) * v))
print(latex(b))
def test_find_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)
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_1(expr))
# ...
# ...
equation = find(u, forall=v, lhs=a1(u,v), rhs=l1(v))
# ...
# ...
a = BilinearForm((v,u), a1(v,u) + alpha*a2(v,u))
equation = find(u, forall=v, lhs=a(u,v), rhs=l1(v))
# ...
# ...
a = BilinearForm((v,u), a1(v,u) + a_B1(v,u))
equation = find(u, forall=v, lhs=a(u,v), rhs=l1(v))
# ...
# ...
a = BilinearForm((v,u), a1(v,u) + a_B1(v,u))
l = LinearForm(v, l1(v) + l2(v))
equation = find(u, forall=v, lhs=a(u,v), rhs=l(v))
# ...
# ...
a = BilinearForm((v,u), a1(v,u) + a_B1(v,u))
l = LinearForm(v, l1(v) + alpha*l_B2(v))
equation = find(u, forall=v, lhs=a(u,v), rhs=l(v))
# ...
# ... using bc
equation = find(u, forall=v, lhs=a(u,v), rhs=l(v), bc=EssentialBC(u, 0, B1))
def test_logical_expr_2d_2():
dim = 2
A = Square('A')
B = Square('B')
M1 = Mapping('M1', dim=dim)
M2 = Mapping('M2', dim=dim)
D1 = M1(A)
D2 = M2(B)
domain = D1.join(D2,
name='domain',
bnd_minus=D1.get_boundary(axis=0, ext=1),
bnd_plus=D2.get_boundary(axis=0, ext=-1))
V1 = ScalarFunctionSpace('V1', domain, kind='h1')
V2 = VectorFunctionSpace('V2', domain, kind='h1')
u1, v1 = [element_of(V1, name=i) for i in ['u1', 'v1']]
u2, v2 = [element_of(V2, name=i) for i in ['u2', 'v2']]
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(domain.interfaces, expr)
expr = LogicalExpr(int_0(u1 * v1), domain)
assert str(
expr.args[0]
) == 'Integral(A, u1*v1*sqrt(det(Jacobian(M1).T * Jacobian(M1))))'
assert str(
expr.args[1]
) == 'Integral(B, u1*v1*sqrt(det(Jacobian(M2).T * Jacobian(M2))))'
expr = LogicalExpr(int_1(plus(u1) * plus(v1)), domain)
assert str(
expr
) == 'Integral(A|B, PlusInterfaceOperator(u1)*PlusInterfaceOperator(v1)*sqrt(det(Jacobian(M1|M2).T * Jacobian(M1|M2))))'
expr = LogicalExpr(int_1(minus(u1) * minus(v1)), domain)
assert str(
expr
) == 'Integral(A|B, MinusInterfaceOperator(u1)*MinusInterfaceOperator(v1)*sqrt(det(Jacobian(M1|M2).T * Jacobian(M1|M2))))'
expr = LogicalExpr(int_0(dot(u2, v2)), domain)
assert str(
expr.args[0]
) == 'Integral(A, Dot(u2, v2)*sqrt(det(Jacobian(M1).T * Jacobian(M1))))'
assert str(
expr.args[1]
) == 'Integral(B, Dot(u2, v2)*sqrt(det(Jacobian(M2).T * Jacobian(M2))))'
expr = LogicalExpr(int_1(dot(plus(u2), plus(v2))), domain)
assert str(
expr
) == 'Integral(A|B, Dot(PlusInterfaceOperator(u2), PlusInterfaceOperator(v2))*sqrt(det(Jacobian(M1|M2).T * Jacobian(M1|M2))))'
expr = LogicalExpr(int_1(dot(minus(u2), minus(v2))), domain)
assert str(
expr
) == 'Integral(A|B, Dot(MinusInterfaceOperator(u2), MinusInterfaceOperator(v2))*sqrt(det(Jacobian(M1|M2).T * Jacobian(M1|M2))))'
expr = LogicalExpr(int_1(dot(grad(minus(u2)), grad(minus(v2)))), domain)
assert str(
expr
) == 'Integral(A|B, Dot((Jacobian(M1)**(-1)).T * Grad(MinusInterfaceOperator(u2)), (Jacobian(M1)**(-1)).T * Grad(MinusInterfaceOperator(v2)))*sqrt(det(Jacobian(M1|M2).T * Jacobian(M1|M2))))'
expr = LogicalExpr(int_1(dot(grad(plus(u2)), grad(plus(v2)))), domain)
assert str(
expr
) == 'Integral(A|B, Dot((Jacobian(M2)**(-1)).T * Grad(PlusInterfaceOperator(u2)), (Jacobian(M2)**(-1)).T * Grad(PlusInterfaceOperator(v2)))*sqrt(det(Jacobian(M1|M2).T * Jacobian(M1|M2))))'
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)
