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_bilinear_expr_2d_2():
domain = Domain('Omega', dim=2)
x,y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu' , is_real=True)
V = VectorFunctionSpace('V', domain)
u,u1,u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v,v1,v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
# ...
a = BilinearExpr((u,v), dot(u,v))
print(a)
print(a.expr)
print(a(u1,v1))
# TODO
# print(a(u1+u2,v1+v2))
print('')
# ...
# ...
a1 = BilinearExpr((u,v), dot(u,v))
a2 = BilinearExpr((u,v), inner(grad(u),grad(v)))
print(a1(u1,v1) + a2(u2,v2))
print('')
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_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_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_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_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_linear_expr_2d_2():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
V = VectorFunctionSpace('V', domain)
u, u1, u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
g = Tuple(x, y)
l = LinearExpr(v, dot(g, v))
print(l)
print(l.expr)
print(l(v1))
# TODO
# print(l(v1+v2))
print('')
# ...
# ...
g1 = Tuple(x, 0)
g2 = Tuple(0, y)
l = LinearExpr((v1, v2), dot(g1, v1) + dot(g2, v2))
print(l)
print(l.expr)
print(l(u1, u2))
# TODO
# print(l(u1+v1, u2+v2))
print('')
def test_terminal_expr_bilinear_2d_2():
domain = Domain('Omega', dim=2)
B1 = Boundary(r'\Gamma_1', domain)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
nn = NormalVector('nn')
V = VectorFunctionSpace('V', domain)
u, u1, u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
# ...
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
a = BilinearForm((u, v), int_0(dot(u, v)))
print(TerminalExpr(a))
print('')
# ...
a = BilinearForm((u, v), int_0(inner(grad(u), grad(v))))
print(TerminalExpr(a))
print('')
# ...
# ...
a = BilinearForm((u, v), int_0(dot(u, v) + inner(grad(u), grad(v))))
print(TerminalExpr(a))
print('')
def test_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_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_calculus_2d_1():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
W = VectorFunctionSpace('W', domain)
alpha, beta, gamma = [Constant(i) for i in ['alpha', 'beta', 'gamma']]
f, g, h = elements_of(V, names='f, g, h')
F, G, H = elements_of(W, names='F, G, H')
# ... scalar gradient properties
assert (grad(f + g) == grad(f) + grad(g))
assert (grad(alpha * h) == alpha * grad(h))
assert (grad(alpha * f + beta * g) == alpha * grad(f) + beta * grad(g))
assert (grad(f * g) == f * grad(g) + g * grad(f))
assert (grad(f / g) == -f * grad(g) / g**2 + grad(f) / g)
assert (expand(grad(f * g * h)) == f * g * grad(h) + f * h * grad(g) +
g * h * grad(f))
# ...
# ... vector gradient properties
assert (grad(F + G) == grad(F) + grad(G))
assert (grad(alpha * H) == alpha * grad(H))
assert (grad(alpha * F + beta * G) == alpha * grad(F) + beta * grad(G))
assert (grad(dot(F, G)) == convect(F, G) + convect(G, F) +
cross(F, curl(G)) - cross(curl(F), G))
# ...
# ... curl properties
assert (curl(f + g) == curl(f) + curl(g))
assert (curl(alpha * h) == alpha * curl(h))
assert (curl(alpha * f + beta * g) == alpha * curl(f) + beta * curl(g))
# ...
# ... laplace properties
assert (laplace(f + g) == laplace(f) + laplace(g))
assert (laplace(alpha * h) == alpha * laplace(h))
assert (laplace(alpha * f + beta * g) == alpha * laplace(f) +
beta * laplace(g))
# ...
# ... divergence properties
assert (div(F + G) == div(F) + div(G))
assert (div(alpha * H) == alpha * div(H))
assert (div(alpha * F + beta * G) == alpha * div(F) + beta * div(G))
assert (div(cross(F, G)) == -dot(F, curl(G)) + dot(G, curl(F)))
# ...
# ... rot properties
assert (rot(F + G) == rot(F) + rot(G))
assert (rot(alpha * H) == alpha * rot(H))
assert (rot(alpha * F + beta * G) == alpha * rot(F) + beta * rot(G))
def test_linearize_expr_2d_1():
domain = Domain('Omega', dim=2)
x,y = domain.coordinates
V1 = ScalarFunctionSpace('V1', domain)
W1 = VectorFunctionSpace('W1', domain)
v1 = element_of(V1, name='v1')
w1 = element_of(W1, name='w1')
alpha = Constant('alpha')
F = element_of(V1, name='F')
G = element_of(W1, 'G')
# ...
l = LinearExpr(v1, F**2*v1)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(v1, dot(grad(F), grad(F))*v1)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(v1, exp(-F)*v1)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(v1, cos(F)*v1)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(v1, cos(F**2)*v1)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(v1, F**2*dot(grad(F), grad(v1)))
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearExpr(w1, dot(rot(G), grad(G))*w1)
a = linearize(l, G, trials='u1')
print(a)
def test_linearize_2d_1():
domain = Domain('Omega', dim=DIM)
x, y = domain.coordinates
V1 = FunctionSpace('V1', domain)
W1 = VectorFunctionSpace('W1', domain)
v1 = TestFunction(V1, name='v1')
w1 = VectorTestFunction(W1, name='w1')
alpha = Constant('alpha')
F = Field('F', space=V1)
G = VectorField(W1, 'G')
# ...
l = LinearForm(v1, F**2 * v1, check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(v1, dot(grad(F), grad(F)) * v1, check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(v1, exp(-F) * v1, check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(v1, cos(F) * v1, check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(v1, cos(F**2) * v1, check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(v1, F**2 * dot(grad(F), grad(v1)), check=True)
a = linearize(l, F, trials='u1')
print(a)
# ...
# ...
l = LinearForm(w1, dot(rot(G), grad(G)) * w1, check=True)
a = linearize(l, G, trials='u1')
print(a)
def test_essential_bc_1():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
W = VectorFunctionSpace('W', domain)
v = element_of(V, name='v')
w = element_of(W, name='w')
B1 = Boundary(r'\Gamma_1', domain)
nn = NormalVector('nn')
# ... scalar case
bc = EssentialBC(v, 0, B1)
assert (bc.variable == v)
assert (bc.order == 0)
assert (bc.normal_component == False)
assert (bc.index_component == None)
# ...
# ... scalar case
bc = EssentialBC(dot(grad(v), nn), 0, B1)
assert (bc.variable == v)
assert (bc.order == 1)
assert (bc.normal_component == False)
assert (bc.index_component == None)
# ...
# ... vector case
bc = EssentialBC(w, 0, B1)
assert (bc.variable == w)
assert (bc.order == 0)
assert (bc.normal_component == False)
assert (bc.index_component == [0, 1])
# ...
# ... vector case
bc = EssentialBC(dot(w, nn), 0, B1)
assert (bc.variable == w)
assert (bc.order == 0)
assert (bc.normal_component == True)
assert (bc.index_component == None)
# ...
# ... vector case
bc = EssentialBC(w[0], 0, B1)
assert (bc.variable == w)
assert (bc.order == 0)
assert (bc.normal_component == False)
assert (bc.index_component == [0])
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_system_2d():
domain = Square()
V = FunctionSpace('V', domain)
x, y = V.coordinates
u, v, p, q = [TestFunction(V, name=i) for i in ['u', 'v', 'p', 'q']]
a1, a2, b1, b2 = [Constant(i, real=True) for i in ['a1', 'a2', 'b1', 'b2']]
# ...
a = BilinearForm((v, u), dot(grad(u), grad(v)))
m = BilinearForm((v, u), u * v)
expr = a(p, u) + a1 * m(p, u) + b1 * m(p, v) + a(
q, v) + a2 * m(q, u) + b2 * m(q, v)
b = BilinearForm(((p, q), (u, v)), expr)
print(evaluate(b, verbose=True))
# ...
# ...
f1 = x * y
f2 = x + y
l1 = LinearForm(p, f1 * p)
l2 = LinearForm(q, f2 * q)
expr = l1(p) + l2(q)
l = LinearForm((p, q), expr)
print(evaluate(l, verbose=True))
def test_calls_2d_3():
domain = Square()
V = FunctionSpace('V', domain)
x, y = domain.coordinates
pn = Field('pn', V)
wn = Field('wn', V)
dp = TestFunction(V, name='dp')
dw = TestFunction(V, name='dw')
tau = TestFunction(V, name='tau')
sigma = TestFunction(V, name='sigma')
Re = Constant('Re', real=True)
dt = Constant('dt', real=True)
alpha = Constant('alpha', real=True)
l1 = LinearForm(tau,
bracket(pn, wn) * tau - 1. / Re * dot(grad(tau), grad(wn)))
# ...
l = LinearForm((tau, sigma), dt * l1(tau))
print(evaluate(l, verbose=True))
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_bilinear_form_2d_4():
domain = Domain('Omega', dim=2)
B1 = Boundary(r'\Gamma_1', domain)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
V = VectorFunctionSpace('V', domain)
u, u1, u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
# ...
a = BilinearForm((u, v), int_0(dot(u, v)))
assert (a.is_symmetric)
# ...
# ...
a = BilinearForm((u, v), int_0(inner(grad(u), grad(v))))
assert (a.is_symmetric)
def test_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 = FunctionSpace('V', domain)
u, v = [TestFunction(V, name=i) for i in ['u', 'v']]
# ...
expr = dot(grad(u), grad(v)) + f(x, y) * u * v
a = BilinearForm((v, u), expr)
print(a)
print(evaluate(a, verbose=True))
print('')
# ...
# ...
expr = f(x, y) * v
l = LinearForm(v, expr)
print(l)
print(evaluate(l, verbose=True))
print('')
def test_latex_2d_3():
DIM = 2
domain = Domain('Omega', dim=DIM)
B1 = Boundary(r'\Gamma_1', domain)
B2 = Boundary(r'\Gamma_2', domain)
B3 = Boundary(r'\Gamma_3', domain)
V = ScalarFunctionSpace('V', domain)
x = V.coordinates
v = element_of(V, name='v')
u = element_of(V, name='u')
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
# ...
expr = dot(grad(v), grad(u))
a_0 = BilinearForm((v, u), int_0(expr))
expr = v * u
a_bnd = BilinearForm((v, u), int_1(expr))
expr = a_0(v, u) + a_bnd(v, u)
a = BilinearForm((v, u), expr)
print(latex(a_0))
print(latex(a_bnd))
print(latex(a))
# print(a)
print('')
def test_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_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_functional_2d_1():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
V = ScalarFunctionSpace('V', domain)
F = element_of(V, name='F')
int_0 = lambda expr: integral(domain, expr)
# ...
expr = x * y
a = Functional(int_0(expr), domain)
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
expr = F - cos(2 * pi * x) * cos(3 * pi * y)
expr = dot(grad(expr), grad(expr))
a = Functional(int_0(expr), domain)
print(a)
print(TerminalExpr(a))
print('')
def test_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_tensorize_2d_3():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
u, v = elements_of(V, names='u,v')
bx = Constant('bx')
by = Constant('by')
b = Tuple(bx, by)
expr = integral(domain, dot(b, grad(v)) * dot(b, grad(u)))
a = BilinearForm((u, v), expr)
print(TensorExpr(a))
print('')
def test_bilinear_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']]
# ...
d_forms = {}
d_forms['a1'] = BilinearForm((u, v), u * v)
d_forms['a2'] = BilinearForm((u, v), u * v + dot(grad(u), grad(v)))
d_forms['a3'] = BilinearForm((u, v), v * trace_1(grad(u), B1))
# Poisson with Nitsch method
a0 = BilinearForm((u, v), dot(grad(u), grad(v)))
a_B1 = BilinearForm(
(u, v), -kappa * u * trace_1(grad(v), B1) - v * trace_1(grad(u), B1) +
trace_0(u, B1) * trace_0(v, B1) / eps)
a = BilinearForm((u, v), a0(u, v) + a_B1(u, v))
d_forms['a4'] = a
# ...
# ... calls
d_calls = {}
for name, a in d_forms.items():
d_calls[name] = a(u1, v1)
# ...
# ... export forms
for name, expr in d_forms.items():
export(expr, 'biform_2d_{}.png'.format(name))
# ...
# ... export calls
for name, expr in d_calls.items():
export(expr, 'biform_2d_call_{}.png'.format(name))
def test_tensorize_3d():
V = FunctionSpace('V', domain)
U = FunctionSpace('U', domain)
W1 = VectorFunctionSpace('W1', domain)
T1 = VectorFunctionSpace('T1', domain)
v = TestFunction(V, name='v')
u = TestFunction(U, name='u')
w1 = VectorTestFunction(W1, name='w1')
t1 = VectorTestFunction(T1, name='t1')
x,y,z = domain.coordinates
alpha = Constant('alpha')
# ...
expr = dot(grad(v), grad(u))
a = BilinearForm((v,u), expr, name='a')
print(a)
print(tensorize(a))
print('')
# ...
# ...
expr = x*dx(v)*dx(u) + y*dy(v)*dy(u)
a = BilinearForm((v,u), expr, name='a')
print(a)
print(tensorize(a))
print('')
# ...
# ...
expr = sin(x)*dx(v)*dx(u)
a = BilinearForm((v,u), expr, name='a')
print(a)
print(tensorize(a))
print('')
# ...
# ...
expr = dot(curl(w1), curl(t1)) + div(w1)*div(t1)
a = BilinearForm((w1, t1), expr, name='a')
print(a)
print(tensorize(a))
print('')
def test_bilinearity_2d_1():
domain = Square()
x, y = domain.coordinates
alpha = Constant('alpha')
beta = Constant('beta')
f1 = x * y
f2 = x + y
f = Tuple(f1, f2)
V = FunctionSpace('V', domain)
# TODO improve: naming are not given the same way
G = Field('G', V)
p, q = [TestFunction(V, name=i) for i in ['p', 'q']]
#####################################
# linear expressions
#####################################
# ...
expr = p * q
assert (is_bilinear_form(expr, (p, q)))
# ...
# ...
expr = dot(grad(p), grad(q))
assert (is_bilinear_form(expr, (p, q)))
# ...
# ...
expr = alpha * dot(grad(p),
grad(q)) + beta * p * q + laplace(p) * laplace(q)
assert (is_bilinear_form(expr, (p, q)))
# ...
#####################################
#####################################
# nonlinear expressions
#####################################
# ...
with pytest.raises(UnconsistentLinearExpressionError):
expr = alpha * dot(grad(p**2), grad(q)) + beta * p * q
is_bilinear_form(expr, (p, q))