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_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_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_mapping_2d_2():
print('============ test_mapping_2d_2 ==============')
rdim = 2
F = Mapping('F', rdim)
domain = Domain('Omega', dim=rdim)
D = F(domain)
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_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_logical_expr_2d_1():
rdim = 2
M = Mapping('M', rdim)
domain = M(Domain('Omega', dim=rdim))
alpha = Constant('alpha')
V = ScalarFunctionSpace('V', domain, kind='h1')
W = VectorFunctionSpace('V', domain, kind='h1')
u, v = [element_of(V, name=i) for i in ['u', 'v']]
w = element_of(W, name='w')
det_M = Jacobian(M).det()
#print('det = ', det_M)
det = Symbol('det')
# ...
expr = 2 * u + alpha * v
expr = LogicalExpr(expr, mapping=M, dim=rdim)
#print(expr)
#print('')
# ...
# ...
expr = dx(u)
expr = LogicalExpr(expr, mapping=M, dim=rdim)
#print(expr.subs(det_M, det))
#print('')
# ...
# ...
expr = dy(u)
expr = LogicalExpr(expr, mapping=M, dim=rdim)
#print(expr.subs(det_M, det))
#print('')
# ...
# ...
expr = dx(det_M)
expr = LogicalExpr(expr, mapping=M, dim=rdim)
expr = expr.subs(det_M, det)
expr = expand(expr)
#print(expr)
#print('')
# ...
# ...
expr = dx(dx(u))
expr = LogicalExpr(expr, mapping=M, dim=rdim)
#print(expr.subs(det_M, det))
#print('')
# ...
# ...
expr = dx(w[0])
expr = LogicalExpr(expr, mapping=M, dim=rdim)
def test_partial_derivatives_2():
print('============ test_partial_derivatives_2 ==============')
# ...
domain = Domain('Omega', dim=2)
M = Mapping('M', dim=2)
mapped_domain = M(domain)
V = ScalarFunctionSpace('V', mapped_domain)
F = element_of(V, name='F')
alpha = Constant('alpha')
beta = Constant('beta')
# ...
# ...
expr = alpha * dx(F)
indices = get_index_derivatives_atom(expr, F)[0]
assert (indices_as_str(indices) == 'x')
# ...
# ...
expr = dy(dx(F))
indices = get_index_derivatives_atom(expr, F)[0]
assert (indices_as_str(indices) == 'xy')
# ...
# ...
expr = alpha * dx(dy(dx(F)))
indices = get_index_derivatives_atom(expr, F)[0]
assert (indices_as_str(indices) == 'xxy')
# ...
# ...
expr = alpha * dx(dx(F)) + beta * dy(F) + dx(dy(F))
indices = get_index_derivatives_atom(expr, F)
indices = [indices_as_str(i) for i in indices]
assert (sorted(indices) == ['xx', 'xy', 'y'])
# ...
# ...
expr = alpha * dx(dx(F)) + beta * dy(F) + dx(dy(F))
d = get_max_partial_derivatives(expr, F)
assert (indices_as_str(d) == 'xxy')
d = get_max_partial_derivatives(expr)
assert (indices_as_str(d) == 'xxy')
def test_logical_expr_3d_1():
rdim = 3
M = Mapping('M', rdim)
domain = Domain('Omega', dim=rdim)
alpha = Constant('alpha')
V = ScalarFunctionSpace('V', domain)
u, v = [element_of(V, name=i) for i in ['u', 'v']]
det_M = DetJacobian(M)
#print('det = ', det_M)
det = Symbol('det')
# ...
expr = 2 * u + alpha * v
expr = LogicalExpr(M, expr)
#print(expr)
#print('')
# ...
# ...
expr = dx(u)
expr = LogicalExpr(M, expr)
#print(expr.subs(det_M, det))
#print('')
# ...
# ...
expr = dy(u)
expr = LogicalExpr(M, expr)
#print(expr.subs(det_M, det))
#print('')
# ...
# ...
expr = dx(det_M)
expr = LogicalExpr(M, expr)
expr = expr.subs(det_M, det)
expr = expand(expr)
#print(expr)
#print('')
# ...
# ...
expr = dx(dx(u))
expr = LogicalExpr(M, expr)
def test_logical_expr_3d_1():
dim = 3
M = Mapping('M', dim=dim)
domain = M(Domain('Omega', dim=dim))
alpha = Constant('alpha')
V = ScalarFunctionSpace('V', domain, kind='h1')
u, v = [element_of(V, name=i) for i in ['u', 'v']]
det_M = Jacobian(M).det()
#print('det = ', det_M)
det = Symbol('det')
# ...
expr = 2 * u + alpha * v
expr = LogicalExpr(expr, domain)
#print(expr)
#print('')
# ...
# ...
expr = dx(u)
expr = LogicalExpr(expr, domain)
#print(expr.subs(det_M, det))
#print('')
# ...
# ...
expr = dy(u)
expr = LogicalExpr(expr, domain)
#print(expr.subs(det_M, det))
#print('')
# ...
# ...
expr = dx(det_M)
expr = LogicalExpr(expr, domain)
expr = expr.subs(det_M, det)
#print(expr)
#print('')
# ...
# ...
expr = dx(dx(u))
expr = LogicalExpr(expr, domain)
def test_tensorize_2d_2_mapping():
DIM = 2
M = Mapping('M', DIM)
domain = Domain('Omega', dim=DIM)
V = VectorFunctionSpace('V', domain)
u, v = elements_of(V, names='u, v')
c = Constant('c')
int_0 = lambda expr: integral(domain, expr)
a = BilinearForm((u, v), int_0(c * div(v) * div(u) + curl(v) * curl(u)))
expr = TensorExpr(a, mapping=M)
print(expr)
def test_expr_mapping_2d():
F = Mapping('F', DIM)
patch = Domain('Omega', dim=DIM)
domain = F(patch)
V = FunctionSpace('V', domain)
v = TestFunction(V, name='v')
u = TestFunction(V, name='u')
x, y = V.coordinates
a = BilinearForm((v, u), dot(grad(v), grad(u)))
assert (a.mapping is F)
l = LinearForm(v, x * y * v)
assert (l.mapping is F)
def test_mapping_1d():
print('============ test_mapping_1d ==============')
rdim = 1
F = Mapping('F', rdim)
assert(F.name == 'F')
# ...
expected = Matrix([[dx1(F[0])]])
assert(F.jacobian == expected)
# ...
# ...
expected = dx1(F[0])
assert(F.det_jacobian == expected)
def test_mapping_1d():
print('============ test_mapping_1d ==============')
dim = 1
F = Mapping('F', dim=dim)
assert(F.name == 'F')
# ...
expected = Matrix([[dx1(F[0])]])
assert(Jacobian(F) == expected)
# ...
# ...
expected = dx1(F[0])
assert(Jacobian(F).det() == expected)
def test_logical_expr_3d_2():
dim = 3
domain = Domain('Omega', dim=dim)
M = Mapping('M', dim=dim)
mapped_domain = M(domain)
V = ScalarFunctionSpace('V', domain, kind='h1')
VM = ScalarFunctionSpace('VM', mapped_domain, kind='h1')
u, v = elements_of(V, names='u,v')
um, vm = elements_of(VM, names='u,v')
J = M.jacobian
a = dot(grad(um), grad(vm))
e = LogicalExpr(a, mapping=M, dim=dim)
assert e == dot(J.inv().T * grad(u), J.inv().T * grad(v))
def test_logical_expr_3d_3():
dim = 3
domain = Domain('Omega', dim=dim)
M = Mapping('M', dim=3)
mapped_domain = M(domain)
V = VectorFunctionSpace('V', domain, kind='hcurl')
VM = VectorFunctionSpace('VM', mapped_domain, kind='hcurl')
u, v = elements_of(V, names='u,v')
um, vm = elements_of(VM, names='u,v')
J = M.jacobian
a = dot(curl(um), curl(vm))
e = LogicalExpr(a, mapping=M, dim=dim)
assert e == dot(J / J.det() * curl(u), J / J.det() * curl(v))
def test_logical_expr_3d_4():
dim = 3
domain = Domain('Omega', dim=dim)
M = Mapping('M', dim=3)
mapped_domain = M(domain)
V = VectorFunctionSpace('V', domain, kind='hdiv')
VM = VectorFunctionSpace('VM', mapped_domain, kind='hdiv')
u, v = elements_of(V, names='u,v')
um, vm = elements_of(VM, names='u,v')
J = M.jacobian
a = div(um) * div(vm)
e = LogicalExpr(a, mapping=M, dim=dim)
assert e == J.det()**-2 * div(u) * div(v)
def test_mapping_3d():
print('============ test_mapping_3d ==============')
rdim = 3
F = Mapping('F', rdim)
a,b,c = symbols('a b c')
abc = Tuple(a, b, c)
assert(F.name == 'F')
# ...
expected = Matrix([[dx1(F[0]), dx2(F[0]), dx3(F[0])],
[dx1(F[1]), dx2(F[1]), dx3(F[1])],
[dx1(F[2]), dx2(F[2]), dx3(F[2])]])
assert(Jacobian(F) == expected)
# ...
# ...
expected = (dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) -
dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) +
dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0]))
assert(Jacobian(F).det() == expected)
# ...
# ...
expected = Tuple (a*(dx2(F[1])*dx3(F[2]) - dx2(F[2])*dx3(F[1]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*(-dx1(F[1])*dx3(F[2]) + dx1(F[2])*dx3(F[1]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*(dx1(F[1])*dx2(F[2]) - dx1(F[2])*dx2(F[1]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])), a*(-dx2(F[0])*dx3(F[2]) + dx2(F[2])*dx3(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*(dx1(F[0])*dx3(F[2]) - dx1(F[2])*dx3(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*(-dx1(F[0])*dx2(F[2]) + dx1(F[2])*dx2(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])), a*(dx2(F[0])*dx3(F[1]) - dx2(F[1])*dx3(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*(-dx1(F[0])*dx3(F[1]) + dx1(F[1])*dx3(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*(dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])))
cov = Covariant(F, abc)
cov = Matrix(cov)
expected = Matrix(expected)
diff = cov-expected
diff.simplify()
assert(diff.dot(diff).is_zero)
# ...
# ...
expected = Tuple (a*dx1(F[0])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*dx2(F[0])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*dx3(F[0])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])), a*dx1(F[1])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*dx2(F[1])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*dx3(F[1])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])), a*dx1(F[2])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*dx2(F[2])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*dx3(F[2])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])))
cov = Contravariant(F, abc)
assert(simplify(cov) == simplify(expected))
def test_topology_1():
# ... create a domain with 2 subdomains A and B
A = InteriorDomain('A', dim=2)
B = InteriorDomain('B', dim=2)
connectivity = Connectivity()
bnd_A_1 = Boundary('Gamma_1', A)
bnd_A_2 = Boundary('Gamma_2', A)
bnd_A_3 = Boundary('Gamma_3', A)
bnd_B_1 = Boundary('Gamma_1', B)
bnd_B_2 = Boundary('Gamma_2', B)
bnd_B_3 = Boundary('Gamma_3', B)
connectivity['I'] = Interface('I', bnd_A_1, bnd_B_2)
mapping = Mapping('M', dim=2)
logical_domain = Domain('D', dim=2)
interiors = [A, B]
boundaries = [bnd_A_2, bnd_A_3, bnd_B_1, bnd_B_3]
Omega = Domain('Omega',
interiors=interiors,
boundaries=boundaries,
connectivity=connectivity,
mapping=mapping,
logical_domain=logical_domain)
interfaces = Omega.interfaces
assert(isinstance(interfaces, Interface))
# export
Omega.export('omega.h5')
# ...
# read it again and check that it has the same description as Omega
D = Domain.from_file('omega.h5')
assert( D.todict() == Omega.todict() )
def test_partial_derivatives_1():
print('============ test_partial_derivatives_1 ==============')
# ...
domain = Domain('Omega', dim=2)
M = Mapping('M', dim=2)
mapped_domain = M(domain)
x, y = mapped_domain.coordinates
V = ScalarFunctionSpace('V', mapped_domain)
F, u, v, w = [element_of(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(y**2) == 0)
assert (dy(y**2) == 2 * y)
assert (dz(y**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_tensorize_2d_1_mapping():
DIM = 2
M = Mapping('Map', dim=DIM)
domain = M(Domain('Omega', dim=DIM))
# 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, domain=domain)
print(expr)
def test_mapping_2d():
print('============ test_mapping_2d ==============')
rdim = 2
F = Mapping('F', rdim)
a,b = symbols('a b')
ab = Tuple(a, b)
assert(F.name == 'F')
# ...
expected = Matrix([[dx1(F[0]), dx2(F[0])],
[dx1(F[1]), dx2(F[1])]])
assert(F.jacobian == expected)
# ...
# ...
expected = dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0])
assert(F.det_jacobian == expected)
# ...
# ...
expected = Tuple(a*dx2(F[1])/(dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0]))
- b*dx1(F[1])/(dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0])),
- a*dx2(F[0])/(dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0]))
+ b*dx1(F[0])/(dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0])))
assert(Covariant(F, ab) == expected)
# ...
# ...
expected = Tuple(a*dx1(F[0])/(dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0]))
+ b*dx2(F[0])/(dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0])),
a*dx1(F[1])/(dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0]))
+ b*dx2(F[1])/(dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0])))
assert(simplify(Contravariant(F, ab)) == simplify(expected))
def test_mapping_3d():
print('============ test_mapping_3d ==============')
rdim = 3
F = Mapping('F', rdim)
a,b,c = symbols('a b c')
abc = Tuple(a, b, c)
assert(F.name == 'F')
# ...
expected = Matrix([[dx1(F[0]), dx2(F[0]), dx3(F[0])],
[dx1(F[1]), dx2(F[1]), dx3(F[1])],
[dx1(F[2]), dx2(F[2]), dx3(F[2])]])
assert(F.jacobian == expected)
# ...
# ...
expected = (dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) -
dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) +
dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0]))
assert(F.det_jacobian == expected)
# ...
# ...
expected = Tuple (a*(dx2(F[1])*dx3(F[2]) - dx2(F[2])*dx3(F[1]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*(-dx1(F[1])*dx3(F[2]) + dx1(F[2])*dx3(F[1]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*(dx1(F[1])*dx2(F[2]) - dx1(F[2])*dx2(F[1]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])), a*(-dx2(F[0])*dx3(F[2]) + dx2(F[2])*dx3(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*(dx1(F[0])*dx3(F[2]) - dx1(F[2])*dx3(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*(-dx1(F[0])*dx2(F[2]) + dx1(F[2])*dx2(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])), a*(dx2(F[0])*dx3(F[1]) - dx2(F[1])*dx3(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*(-dx1(F[0])*dx3(F[1]) + dx1(F[1])*dx3(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*(dx1(F[0])*dx2(F[1]) - dx1(F[1])*dx2(F[0]))/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])))
cov = Covariant(F, abc)
assert(simplify(cov) == simplify(expected))
# ...
# ...
expected = Tuple (a*dx1(F[0])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*dx2(F[0])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*dx3(F[0])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])), a*dx1(F[1])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*dx2(F[1])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*dx3(F[1])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])), a*dx1(F[2])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + b*dx2(F[2])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])) + c*dx3(F[2])/(dx1(F[0])*dx2(F[1])*dx3(F[2]) - dx1(F[0])*dx2(F[2])*dx3(F[1]) - dx1(F[1])*dx2(F[0])*dx3(F[2]) + dx1(F[1])*dx2(F[2])*dx3(F[0]) + dx1(F[2])*dx2(F[0])*dx3(F[1]) - dx1(F[2])*dx2(F[1])*dx3(F[0])))
cov = Contravariant(F, abc)
assert(simplify(cov) == simplify(expected))
def test_symbolic_expr_1d_1():
rdim = 1
M = Mapping('M', rdim)
domain = M(Domain('Omega', dim=rdim))
alpha = Constant('alpha')
V = ScalarFunctionSpace('V', domain, kind='h1')
u = element_of(V, name='u')
det_M = Jacobian(M).det()
det_M = SymbolicExpr(det_M)
#print('>>> ', det_M)
det = Symbol('det')
# ...
expr = u
expr = LogicalExpr(expr, mapping=M, dim=rdim)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx1(u)
expr = LogicalExpr(expr, mapping=M, dim=rdim)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx1(M[0])
expr = LogicalExpr(expr, mapping=M, dim=rdim)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx(u)
expr = LogicalExpr(expr, mapping=M, dim=rdim)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
#print(expr)
# ...
# ...
expr = dx(Jacobian(M).det())
expr = LogicalExpr(expr, mapping=M, dim=rdim)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
#print(expand(expr))
# ...
# ...
expr = dx(dx(u))
expr = LogicalExpr(expr, mapping=M, dim=rdim)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
#print(expand(expr))
# ...
# ...
expr = dx(dx(dx(u)))
expr = LogicalExpr(expr, mapping=M, dim=rdim)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
from nodes import StencilMatrixLocalBasis
from nodes import StencilVectorLocalBasis
from nodes import StencilMatrixGlobalBasis
from nodes import StencilVectorGlobalBasis
from nodes import ElementOf
from nodes import WeightedVolumeQuadrature
from nodes import ComputeKernelExpr
from nodes import AST
from nodes import Block
from nodes import Reset
from parser import parse
# ... abstract model
domain = Square()
M = Mapping('M', domain.dim)
V = ScalarFunctionSpace('V', domain)
u,v = elements_of(V, names='u,v')
expr = dot(grad(v), grad(u))
# ...
# ...
grid = Grid()
element = Element()
g_quad = GlobalTensorQuadrature()
l_quad = LocalTensorQuadrature()
quad = TensorQuadrature()
g_basis = GlobalTensorQuadratureTrialBasis(u)
l_basis = LocalTensorQuadratureTrialBasis(u)
def test_symbolic_expr_3d_1():
rdim = 3
M = Mapping('M', rdim)
domain = Domain('Omega', dim=rdim)
alpha = Constant('alpha')
V = ScalarFunctionSpace('V', domain)
u = element_of(V, 'u')
det_M = DetJacobian(M)
det_M = SymbolicExpr(det_M)
#print('>>> ', det_M)
det = Symbol('det')
# ...
expr = u
expr = LogicalExpr(M, expr)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx1(u)
expr = LogicalExpr(M, expr)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx1(dx2(u))
expr = LogicalExpr(M, expr)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx1(M[0])
expr = LogicalExpr(M, expr)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx(u)
expr = LogicalExpr(M, expr)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
#print(expr)
# ...
# ...
expr = dx(DetJacobian(M))
expr = LogicalExpr(M, expr)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
#print(expand(expr))
# ...
# ...
expr = dx(dx(u))
expr = LogicalExpr(M, expr)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
#print(expand(expr))
# ...
# ...
expr = dx(dx(dx(u)))
expr = LogicalExpr(M, expr)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
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_symbolic_expr_3d_1():
dim = 3
M = Mapping('M', dim=dim)
domain = M(Domain('Omega', dim=dim))
V = ScalarFunctionSpace('V', domain, kind='h1')
u = element_of(V, 'u')
det_M = Jacobian(M).det()
det_M = SymbolicExpr(det_M)
#print('>>> ', det_M)
det = Symbol('det')
# ...
expr = u
expr = LogicalExpr(expr, domain)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx1(u)
expr = LogicalExpr(expr, domain)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx1(dx2(u))
expr = LogicalExpr(expr, domain)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx1(M[0])
expr = LogicalExpr(expr, domain)
expr = SymbolicExpr(expr)
#print(expr)
# ...
# ...
expr = dx(u)
expr = LogicalExpr(expr, domain)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
#print(expr)
# ...
# ...
expr = dx(Jacobian(M).det())
expr = LogicalExpr(expr, domain)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
#print(expand(expr))
# ...
# ...
expr = dx(dx(u))
expr = LogicalExpr(expr, domain)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
#print(expand(expr))
# ...
# ...
expr = dx(dx(dx(u)))
expr = LogicalExpr(expr, domain)
expr = SymbolicExpr(expr)
expr = expr.subs(det_M, det)
