def test_target_mapping_2d_1():
rdim = 2
x1, x2 = symbols('x1, x2')
constants = ['c1', 'c2', 'D', 'k']
c1, c2, D, k = [Constant(i) for i in constants]
M = TargetMapping('M', rdim)
assert (not (M[0] == x1))
assert (not (M[1] == x2))
assert (LogicalExpr(M, M[0]) == -D * x1**2 + c1 + x1 * (-k + 1) * cos(x2))
assert (LogicalExpr(M, M[1]) == c2 + x1 * (k + 1) * sin(x2))
assert (LogicalExpr(M, dx1(M[0])) == -2 * D * x1 - k * x1 * cos(x2) +
(-k + 1) * cos(x2))
assert (LogicalExpr(M, dx1(M[1])) == (k + 1) * sin(x2))
assert (LogicalExpr(M,
dx2(M[0])) == x1 * (-k * cos(x2) - (-k + 1) * sin(x2)))
assert (LogicalExpr(M, dx2(M[1])) == x1 * (k + 1) * cos(x2))
expected = Matrix([[
-2 * D * x1 - k * x1 * cos(x2) + (-k + 1) * cos(x2),
x1 * (-k * cos(x2) - (-k + 1) * sin(x2))
], [(k + 1) * sin(x2), x1 * (k + 1) * cos(x2)]])
assert (not (M.jacobian == expected))
assert (expand(LogicalExpr(M, M.jacobian)) == expand(expected))
def test_polar_mapping_2d_1():
rdim = 2
x1, x2 = symbols('x1, x2')
constants = ['c1', 'c2', 'rmax', 'rmin']
c1, c2, rmax, rmin = [Constant(i) for i in constants]
M = PolarMapping('M', rdim)
assert (not (M[0] == x1))
assert (not (M[1] == x2))
assert (LogicalExpr(M[0], mapping=M, dim=rdim, subs=True) == c1 +
(rmax * x1 + rmin * (-x1 + 1)) * cos(x2))
assert (LogicalExpr(M[1], mapping=M, dim=rdim, subs=True) == c2 +
(rmax * x1 + rmin * (-x1 + 1)) * sin(x2))
assert (LogicalExpr(dx1(M[0]), mapping=M, dim=rdim,
subs=True) == (rmax - rmin) * cos(x2))
assert (LogicalExpr(dx1(M[1]), mapping=M, dim=rdim,
subs=True) == (rmax - rmin) * sin(x2))
expected = -(rmax * x1 + rmin * (-x1 + 1)) * sin(x2)
assert (expand(LogicalExpr(dx2(M[0]), mapping=M, dim=rdim,
subs=True)) == expand(expected))
assert (LogicalExpr(dx2(M[1]), mapping=M, dim=rdim,
subs=True) == (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), mapping=M, dim=rdim,
subs=True)) == expand(expected))
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_target_mapping_2d_1():
rdim = 2
x1, x2 = symbols('x1, x2')
constants = ['c1', 'c2', 'D', 'k']
c1, c2, D, k = [Constant(i) for i in constants]
M = TargetMapping('M', rdim)
assert (not (M[0] == x1))
assert (not (M[1] == x2))
assert (LogicalExpr(M[0], mapping=M, dim=rdim, subs=True) == -D * x1**2 +
c1 + x1 * (-k + 1) * cos(x2))
assert (LogicalExpr(M[1], mapping=M, dim=rdim,
subs=True) == c2 + x1 * (k + 1) * sin(x2))
assert (LogicalExpr(dx1(M[0]), mapping=M, dim=rdim,
subs=True) == -2 * D * x1 + (-k + 1) * cos(x2))
assert (LogicalExpr(dx1(M[1]), mapping=M, dim=rdim,
subs=True) == (k + 1) * sin(x2))
assert (LogicalExpr(dx2(M[0]), mapping=M, dim=rdim,
subs=True) == -x1 * (-k + 1) * sin(x2))
assert (LogicalExpr(dx2(M[1]), mapping=M, dim=rdim,
subs=True) == 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 (not (Jacobian(M) == expected))
assert (expand(LogicalExpr(Jacobian(M), mapping=M, dim=rdim,
subs=True)) == expand(expected))
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_twisted_target_mapping_3d_1():
rdim = 3
x1, x2, x3 = symbols('x1, x2, x3')
constants = ['c1', 'c2', 'c3', 'D', 'k']
c1, c2, c3, D, k = [Constant(i) for i in constants]
M = TwistedTargetMapping('M', rdim)
assert (not (M[0] == x1))
assert (not (M[1] == x2))
assert (not (M[2] == x3))
expected = -D * x1**2 + c1 + x1 * (-k + 1) * cos(x2)
assert (LogicalExpr(M[0], mapping=M, dim=rdim, subs=True) == expected)
expected = c2 + x1 * (k + 1) * sin(x2)
assert (LogicalExpr(M[1], mapping=M, dim=rdim, subs=True) == expected)
expected = c3 + x1**2 * x3 * sin(2 * x2)
assert (LogicalExpr(M[2], mapping=M, dim=rdim, subs=True) == expected)
expected = -2 * D * x1 + (-k + 1) * cos(x2)
assert (LogicalExpr(dx1(M[0]), mapping=M, dim=rdim, subs=True) == expected)
expected = (k + 1) * sin(x2)
assert (LogicalExpr(dx1(M[1]), mapping=M, dim=rdim, subs=True) == expected)
expected = 2 * x1 * x3 * sin(2 * x2)
assert (LogicalExpr(dx1(M[2]), mapping=M, dim=rdim, subs=True) == expected)
expected = -x1 * (-k + 1) * sin(x2)
assert (LogicalExpr(dx2(M[0]), mapping=M, dim=rdim, subs=True) == expected)
expected = x1 * (k + 1) * cos(x2)
assert (LogicalExpr(dx2(M[1]), mapping=M, dim=rdim, subs=True) == expected)
expected = 2 * x1**2 * x3 * cos(2 * x2)
assert (LogicalExpr(dx2(M[2]), mapping=M, dim=rdim, subs=True) == expected)
expected = 0
assert (expand(LogicalExpr(dx3(M[0]), mapping=M, dim=rdim,
subs=True)) == expand(expected))
expected = 0
assert (LogicalExpr(dx3(M[1]), mapping=M, dim=rdim, subs=True) == expected)
expected = x1**2 * sin(2 * x2)
assert (LogicalExpr(dx3(M[2]), mapping=M, dim=rdim, subs=True) == expected)
expected = Matrix(
[[-2 * D * x1 + (-k + 1) * cos(x2), -x1 * (-k + 1) * sin(x2), 0],
[(k + 1) * sin(x2), x1 * (k + 1) * cos(x2), 0],
[
2 * x1 * x3 * sin(2 * x2), 2 * x1**2 * x3 * cos(2 * x2),
x1**2 * sin(2 * x2)
]])
assert (not (Jacobian(M) == expected))
assert (expand(LogicalExpr(Jacobian(M), mapping=M, dim=rdim,
subs=True)) == expand(expected))
def test_torus_mapping_3d_1():
dim = 3
x1, x2, x3 = symbols('x1, x2, x3')
R0 = Constant('R0')
M = TorusMapping('M', dim=dim)
domain = M(Domain('Omega', dim=dim))
assert (not (M[0] == x1))
assert (not (M[1] == x2))
assert (not (M[2] == x3))
expected = (R0 + x1 * cos(x2)) * cos(x3)
assert (LogicalExpr(M[0], domain) == expected)
expected = (R0 + x1 * cos(x2)) * sin(x3)
assert (LogicalExpr(M[1], domain) == expected)
expected = x1 * sin(x2)
assert (LogicalExpr(M[2], domain) == expected)
expected = cos(x2) * cos(x3)
assert (LogicalExpr(dx1(M[0]), domain) == expected)
expected = sin(x3) * cos(x2)
assert (LogicalExpr(dx1(M[1]), domain) == expected)
expected = sin(x2)
assert (LogicalExpr(dx1(M[2]), domain) == expected)
expected = -x1 * sin(x2) * cos(x3)
assert (LogicalExpr(dx2(M[0]), domain) == expected)
expected = -x1 * sin(x2) * sin(x3)
assert (LogicalExpr(dx2(M[1]), domain) == expected)
expected = x1 * cos(x2)
assert (LogicalExpr(dx2(M[2]), domain) == expected)
expected = -(R0 + x1 * cos(x2)) * sin(x3)
assert (expand(LogicalExpr(dx3(M[0]), domain)) == expand(expected))
expected = (R0 + x1 * cos(x2)) * cos(x3)
assert (LogicalExpr(dx3(M[1]), domain) == expected)
expected = 0
assert (LogicalExpr(dx3(M[2]), domain) == expected)
expected = Matrix([[
cos(x2) * cos(x3), -x1 * sin(x2) * cos(x3),
-(R0 + x1 * cos(x2)) * sin(x3)
],
[
sin(x3) * cos(x2), -x1 * sin(x2) * sin(x3),
(R0 + x1 * cos(x2)) * cos(x3)
], [sin(x2), x1 * cos(x2), 0]])
assert (all(e.expand().is_zero for e in (Jacobian(M) - expected)))
def test_torus_mapping_3d_1():
rdim = 3
x1, x2, x3 = symbols('x1, x2, x3')
R0 = Constant('R0')
M = TorusMapping('M', rdim)
assert (not (M[0] == x1))
assert (not (M[1] == x2))
assert (not (M[2] == x3))
expected = (R0 + x1 * cos(x2)) * cos(x3)
assert (LogicalExpr(M[0], mapping=M, dim=rdim, subs=True) == expected)
expected = (R0 + x1 * cos(x2)) * sin(x3)
assert (LogicalExpr(M[1], mapping=M, dim=rdim, subs=True) == expected)
expected = x1 * sin(x2)
assert (LogicalExpr(M[2], mapping=M, dim=rdim, subs=True) == expected)
expected = cos(x2) * cos(x3)
assert (LogicalExpr(dx1(M[0]), mapping=M, dim=rdim, subs=True) == expected)
expected = sin(x3) * cos(x2)
assert (LogicalExpr(dx1(M[1]), mapping=M, dim=rdim, subs=True) == expected)
expected = sin(x2)
assert (LogicalExpr(dx1(M[2]), mapping=M, dim=rdim, subs=True) == expected)
expected = -x1 * sin(x2) * cos(x3)
assert (LogicalExpr(dx2(M[0]), mapping=M, dim=rdim, subs=True) == expected)
expected = -x1 * sin(x2) * sin(x3)
assert (LogicalExpr(dx2(M[1]), mapping=M, dim=rdim, subs=True) == expected)
expected = x1 * cos(x2)
assert (LogicalExpr(dx2(M[2]), mapping=M, dim=rdim, subs=True) == expected)
expected = -(R0 + x1 * cos(x2)) * sin(x3)
assert (expand(LogicalExpr(dx3(M[0]), mapping=M, dim=rdim,
subs=True)) == expand(expected))
expected = (R0 + x1 * cos(x2)) * cos(x3)
assert (LogicalExpr(dx3(M[1]), mapping=M, dim=rdim, subs=True) == expected)
expected = 0
assert (LogicalExpr(dx3(M[2]), mapping=M, dim=rdim, subs=True) == expected)
expected = Matrix([[
cos(x2) * cos(x3), -x1 * sin(x2) * cos(x3),
-(R0 + x1 * cos(x2)) * sin(x3)
],
[
sin(x3) * cos(x2), -x1 * sin(x2) * sin(x3),
(R0 + x1 * cos(x2)) * cos(x3)
], [sin(x2), x1 * cos(x2), 0]])
assert (not (Jacobian(M) == expected))
assert (expand(LogicalExpr(Jacobian(M), mapping=M, dim=rdim,
subs=True)) == expand(expected))
def test_collela_mapping_2d_1():
dim = 2
x1, x2 = symbols('x1, x2')
constants = ['eps', 'k1', 'k2']
eps, k1, k2 = [Constant(i) for i in constants]
M = CollelaMapping2D('M', dim)
domain = M(Domain('Omega', dim=dim))
assert (not (M[0] == x1))
assert (not (M[1] == x2))
expected = 2.0 * eps * sin(2.0 * k1 * pi * x1) * sin(
2.0 * k2 * pi * x2) + 2.0 * x1 - 1.0
assert (LogicalExpr(M[0], domain) == expected)
expected = 2.0 * eps * sin(2.0 * k1 * pi * x1) * sin(
2.0 * k2 * pi * x2) + 2.0 * x2 - 1.0
assert (LogicalExpr(M[1], domain) == expected)
expected = 4.0 * eps * k1 * pi * sin(2.0 * k2 * pi * x2) * cos(
2.0 * k1 * pi * x1) + 2.0
assert (LogicalExpr(dx1(M[0]), domain) == expected)
expected = 4.0 * eps * k1 * pi * sin(2.0 * k2 * pi * x2) * cos(
2.0 * k1 * pi * x1)
assert (LogicalExpr(dx1(M[1]), domain) == expected)
expected = 4.0 * eps * k2 * pi * sin(2.0 * k1 * pi * x1) * cos(
2.0 * k2 * pi * x2)
assert (LogicalExpr(dx2(M[0]), domain) == expected)
expected = 4.0 * eps * k2 * pi * sin(2.0 * k1 * pi * x1) * cos(
2.0 * k2 * pi * x2) + 2.0
assert (LogicalExpr(dx2(M[1]), domain) == expected)
expected = Matrix([[
4.0 * eps * k1 * pi * sin(2.0 * k2 * pi * x2) * cos(2.0 * k1 * pi * x1)
+ 2.0,
4.0 * eps * k2 * pi * sin(2.0 * k1 * pi * x1) * cos(2.0 * k2 * pi * x2)
],
[
4.0 * eps * k1 * pi * sin(2.0 * k2 * pi * x2) *
cos(2.0 * k1 * pi * x1),
4.0 * eps * k2 * pi * sin(2.0 * k1 * pi * x1) *
cos(2.0 * k2 * pi * x2) + 2.0
]])
assert ((Jacobian(M) == expected))
def test_collela_mapping_2d_1():
rdim = 2
x1, x2 = symbols('x1, x2')
constants = ['eps', 'k1', 'k2']
eps, k1, k2 = [Constant(i) for i in constants]
M = CollelaMapping('M', rdim)
assert (not (M[0] == x1))
assert (not (M[1] == x2))
expected = 2.0 * eps * sin(2.0 * k1 * pi * x1) * sin(
2.0 * k2 * pi * x2) + 2.0 * x1 - 1.0
assert (LogicalExpr(M[0], mapping=M, dim=rdim, subs=True) == expected)
expected = 2.0 * eps * sin(2.0 * k1 * pi * x1) * sin(
2.0 * k2 * pi * x2) + 2.0 * x2 - 1.0
assert (LogicalExpr(M[1], mapping=M, dim=rdim, subs=True) == expected)
expected = 4.0 * eps * k1 * pi * sin(2.0 * k2 * pi * x2) * cos(
2.0 * k1 * pi * x1) + 2.0
assert (LogicalExpr(dx1(M[0]), mapping=M, dim=rdim, subs=True) == expected)
expected = 4.0 * eps * k1 * pi * sin(2.0 * k2 * pi * x2) * cos(
2.0 * k1 * pi * x1)
assert (LogicalExpr(dx1(M[1]), mapping=M, dim=rdim, subs=True) == expected)
expected = 4.0 * eps * k2 * pi * sin(2.0 * k1 * pi * x1) * cos(
2.0 * k2 * pi * x2)
assert (LogicalExpr(dx2(M[0]), mapping=M, dim=rdim, subs=True) == expected)
expected = 4.0 * eps * k2 * pi * sin(2.0 * k1 * pi * x1) * cos(
2.0 * k2 * pi * x2) + 2.0
assert (LogicalExpr(dx2(M[1]), mapping=M, dim=rdim, subs=True) == expected)
expected = Matrix([[
4.0 * eps * k1 * pi * sin(2.0 * k2 * pi * x2) * cos(2.0 * k1 * pi * x1)
+ 2.0,
4.0 * eps * k2 * pi * sin(2.0 * k1 * pi * x1) * cos(2.0 * k2 * pi * x2)
],
[
4.0 * eps * k1 * pi * sin(2.0 * k2 * pi * x2) *
cos(2.0 * k1 * pi * x1),
4.0 * eps * k2 * pi * sin(2.0 * k1 * pi * x1) *
cos(2.0 * k2 * pi * x2) + 2.0
]])
assert (not (Jacobian(M) == expected))
assert (expand(LogicalExpr(Jacobian(M), mapping=M, dim=rdim,
subs=True)) == expand(expected))
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_eval_field_2d_1():
# ...
args = [dx1(u), dx2(u)]
# TODO improve
stmts = [AugAssign(ProductGenerator(MatrixQuadrature(i), index_quad),
'+', Mul(coeff,AtomicNode(i)))
for i in args]
# ...
# ...
nderiv = 1
body = construct_logical_expressions(u, nderiv)
# ...
# ...
stmts = body + stmts
loop = Loop((l_quad, a_basis), index_quad, stmts)
# ...
# ...
stmts = [loop]
loop = Loop((l_basis, l_coeff), index_dof, stmts)
# ...
# TODO do we need nderiv here?
stmt = parse(loop, settings={'dim': domain.dim, 'nderiv': nderiv})
print(pycode(stmt))
print()
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_identity_mapping_2d_2():
dim = 2
M = IdentityMapping('F', dim=dim)
domain = M(Domain('Omega', dim=dim))
V = ScalarFunctionSpace('V', domain, kind='h1')
u = element_of(V, name='u')
# ...
assert (LogicalExpr(dx(u), domain) == dx1(u))
assert (LogicalExpr(dy(u), domain) == dx2(u))
def test_identity_mapping_2d_1():
rdim = 2
x1, x2 = symbols('x1, x2')
M = IdentityMapping('M', rdim)
assert (not (M[0] == x1))
assert (not (M[1] == x2))
assert (LogicalExpr(M, M[0]) == x1)
assert (LogicalExpr(M, M[1]) == x2)
assert (LogicalExpr(M, dx1(M[0])) == 1)
assert (LogicalExpr(M, dx1(M[1])) == 0)
assert (LogicalExpr(M, dx2(M[0])) == 0)
assert (LogicalExpr(M, dx2(M[1])) == 1)
expected = Matrix([[1, 0], [0, 1]])
assert (not (M.jacobian == expected))
assert (LogicalExpr(M, M.jacobian) == expected)
def test_identity_mapping_2d_1():
dim = 2
x1, x2 = symbols('x1, x2')
M = IdentityMapping('M', dim=dim)
domain = M(Domain('Omega', dim=dim))
assert (not (M[0] == x1))
assert (not (M[1] == x2))
assert (LogicalExpr(M[0], domain) == x1)
assert (LogicalExpr(M[1], domain) == x2)
assert (LogicalExpr(dx1(M[0]), domain) == 1)
assert (LogicalExpr(dx1(M[1]), domain) == 0)
assert (LogicalExpr(dx2(M[0]), domain) == 0)
assert (LogicalExpr(dx2(M[1]), domain) == 1)
expected = Matrix([[1, 0], [0, 1]])
assert (Jacobian(M) == expected)
def test_identity_mapping_2d_2():
rdim = 2
x1, x2 = symbols('x1, x2')
domain = Domain('Omega', dim=rdim)
M = IdentityMapping('F', rdim)
V = ScalarFunctionSpace('V', domain)
u = element_of(V, name='u')
# ...
assert (LogicalExpr(M, dx(u)) == dx1(u))
assert (LogicalExpr(M, dy(u)) == dx2(u))
def test_identity_mapping_2d_2():
rdim = 2
x1, x2 = symbols('x1, x2')
M = IdentityMapping('F', rdim)
domain = M(Domain('Omega', dim=rdim))
V = ScalarFunctionSpace('V', domain, kind='h1')
u = element_of(V, name='u')
# ...
assert (LogicalExpr(dx(u), mapping=M, dim=rdim, subs=True) == dx1(u))
assert (LogicalExpr(dy(u), mapping=M, dim=rdim, subs=True) == dx2(u))
def test_loop_local_dof_quad_2d_3():
# ...
stmts = [dx1(u)]
stmts = [ComputeLogicalBasis(i) for i in stmts]
# ...
# ...
loop = Loop((l_quad, a_basis), index_quad, stmts)
# ...
# ...
stmts = [loop]
loop = Loop(l_basis, index_dof, stmts)
# ...
stmt = parse(loop, settings={'dim': domain.dim, 'nderiv': 3})
print()
print(pycode(stmt))
print()
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)
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_czarny_mapping_2d_1():
rdim = 2
x1, x2 = symbols('x1, x2')
constants = ['c2', 'eps', 'b']
c2, eps, b = [Constant(i) for i in constants]
M = CzarnyMapping('M', rdim)
assert (not (M[0] == x1))
assert (not (M[1] == x2))
expected = (-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 1) / eps
assert (LogicalExpr(M[0], mapping=M, dim=rdim, subs=True) == expected)
expected = b * x1 * sin(x2) / (
sqrt(-eps**2 / 4 + 1) * (-sqrt(eps *
(eps + 2 * x1 * cos(x2)) + 1) + 2)) + c2
assert (LogicalExpr(M[1], mapping=M, dim=rdim, subs=True) == expected)
expected = -cos(x2) / sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1)
assert (LogicalExpr(dx1(M[0]), mapping=M, dim=rdim, subs=True) == expected)
expected = b * (
eps * x1 * sin(x2) * cos(x2) /
(sqrt(-eps**2 / 4 + 1) * sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) *
(-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 2)**2) + sin(x2) /
(sqrt(-eps**2 / 4 + 1) * (-sqrt(eps *
(eps + 2 * x1 * cos(x2)) + 1) + 2)))
assert ((LogicalExpr(dx1(M[1]), mapping=M, dim=rdim, subs=True) -
expected).expand() == 0)
expected = x1 * sin(x2) / sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1)
assert (LogicalExpr(dx2(M[0]), mapping=M, dim=rdim, subs=True) == expected)
expected = b * x1 * (
-eps * x1 * sin(x2)**2 /
(sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) *
(-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 2)**2) + cos(x2) /
(-sqrt(eps *
(eps + 2 * x1 * cos(x2)) + 1) + 2)) / sqrt(-eps**2 / 4 + 1)
assert ((LogicalExpr(dx2(M[1]), mapping=M, dim=rdim, subs=True) -
expected).expand() == 0)
expected = Matrix(
[[
-cos(x2) / sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1),
x1 * sin(x2) / sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1)
],
[
b * (eps * x1 * sin(x2) * cos(x2) /
(sqrt(-eps**2 / 4 + 1) * sqrt(eps *
(eps + 2 * x1 * cos(x2)) + 1) *
(-sqrt(eps *
(eps + 2 * x1 * cos(x2)) + 1) + 2)**2) + sin(x2) /
(sqrt(-eps**2 / 4 + 1) *
(-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 2))), b * x1 *
(-eps * x1 * sin(x2)**2 /
(sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) *
(-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 2)**2) + cos(x2) /
(-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 2)) /
sqrt(-eps**2 / 4 + 1)
]])
assert (not (Jacobian(M) == expected))
assert (expand(LogicalExpr(Jacobian(M), mapping=M, dim=rdim,
subs=True)) == expand(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(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_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_czarny_mapping_2d_1():
dim = 2
x1, x2 = symbols('x1, x2')
constants = ['c2', 'eps', 'b']
c2, eps, b = [Constant(i) for i in constants]
M = CzarnyMapping('M', dim=dim)
domain = M(Domain('Omega', dim=dim))
assert (not (M[0] == x1))
assert (not (M[1] == x2))
expected = (-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 1) / eps
assert (LogicalExpr(M[0], domain) == expected)
expected = b * x1 * sin(x2) / (
sqrt(-eps**2 / 4 + 1) * (-sqrt(eps *
(eps + 2 * x1 * cos(x2)) + 1) + 2)) + c2
assert (LogicalExpr(M[1], domain) == expected)
expected = -cos(x2) / sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1)
assert (LogicalExpr(dx1(M[0]), domain) == expected)
expected = b * (
eps * x1 * sin(x2) * cos(x2) /
(sqrt(-eps**2 / 4 + 1) * sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) *
(-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 2)**2) + sin(x2) /
(sqrt(-eps**2 / 4 + 1) * (-sqrt(eps *
(eps + 2 * x1 * cos(x2)) + 1) + 2)))
assert ((LogicalExpr(dx1(M[1]), domain) - expected).expand() == 0)
expected = x1 * sin(x2) / sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1)
assert (LogicalExpr(dx2(M[0]), domain) == expected)
expected = b * x1 * (
-eps * x1 * sin(x2)**2 /
(sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) *
(-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 2)**2) + cos(x2) /
(-sqrt(eps *
(eps + 2 * x1 * cos(x2)) + 1) + 2)) / sqrt(-eps**2 / 4 + 1)
assert ((LogicalExpr(dx2(M[1]), domain) - expected).expand() == 0)
expected = Matrix(
[[
-cos(x2) / sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1),
x1 * sin(x2) / sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1)
],
[
b * (eps * x1 * sin(x2) * cos(x2) /
(sqrt(-eps**2 / 4 + 1) * sqrt(eps *
(eps + 2 * x1 * cos(x2)) + 1) *
(-sqrt(eps *
(eps + 2 * x1 * cos(x2)) + 1) + 2)**2) + sin(x2) /
(sqrt(-eps**2 / 4 + 1) *
(-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 2))), b * x1 *
(-eps * x1 * sin(x2)**2 /
(sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) *
(-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 2)**2) + cos(x2) /
(-sqrt(eps * (eps + 2 * x1 * cos(x2)) + 1) + 2)) /
sqrt(-eps**2 / 4 + 1)
]])
assert (all(e.expand().is_zero for e in (Jacobian(M) - expected)))
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)
