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_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_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_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_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_interface_2d_1():
# ...
def two_patches():
from sympde.topology import InteriorDomain
from sympde.topology import Connectivity, Interface
A = Square('A')
B = Square('B')
A = A.interior
B = B.interior
connectivity = Connectivity()
bnd_A_1 = Boundary(r'\Gamma_1', A, axis=0, ext=-1)
bnd_A_2 = Boundary(r'\Gamma_2', A, axis=0, ext=1)
bnd_A_3 = Boundary(r'\Gamma_3', A, axis=1, ext=-1)
bnd_A_4 = Boundary(r'\Gamma_4', A, axis=1, ext=1)
bnd_B_1 = Boundary(r'\Gamma_1', B, axis=0, ext=-1)
bnd_B_2 = Boundary(r'\Gamma_2', B, axis=0, ext=1)
bnd_B_3 = Boundary(r'\Gamma_3', B, axis=1, ext=-1)
bnd_B_4 = Boundary(r'\Gamma_4', B, axis=1, ext=1)
connectivity['I'] = Interface('I', bnd_A_2, bnd_B_1)
Omega = Domain('Omega',
interiors=[A, B],
boundaries=[
bnd_A_1, bnd_A_2, bnd_A_3, bnd_A_4, bnd_B_1,
bnd_B_2, bnd_B_3, bnd_B_4
],
connectivity=connectivity)
return Omega
# ...
# create a domain with an interface
domain = two_patches()
interfaces = domain.interfaces
V = ScalarFunctionSpace('V', domain)
u, v = elements_of(V, names='u, v')
print(integral(interfaces, u * v))
expr = integral(domain, dot(grad(v), grad(u)))
expr += integral(interfaces, -avg(Dn(u)) * jump(v) + avg(Dn(v)) * jump(u))
a = BilinearForm((u, v), expr)
print(a)
def test_interface_integral_1():
# ...
A = Square('A')
B = Square('B')
domain = A.join(B,
name='domain',
bnd_minus=A.get_boundary(axis=0, ext=1),
bnd_plus=B.get_boundary(axis=0, ext=-1))
# ...
x, y = domain.coordinates
V = ScalarFunctionSpace('V', domain, kind=None)
assert (V.is_broken)
u, v = elements_of(V, names='u, v')
# ...
I = domain.interfaces
# ...
# expr = minus(Dn(u))
# print(expr)
# import sys; sys.exit(0)
# ... bilinear forms
# a = BilinearForm((u,v), integral(domain, u*v))
# a = BilinearForm((u,v), integral(domain, dot(grad(u),grad(v))))
# a = BilinearForm((u,v), integral(I, jump(u) * jump(v)))
# a = BilinearForm((u,v), integral(I, jump(Dn(u)) * jump(v)))
# a = BilinearForm((u,v), integral(domain, dot(grad(u),grad(v)))
# + integral(I, jump(u) * jump(v)))
# Nitsch
kappa = Constant('kappa')
expr_I = (-jump(u) * jump(Dn(v)) + kappa * jump(u) * jump(v) +
plus(Dn(u)) * minus(v) + minus(Dn(u)) * plus(v))
a = BilinearForm(
(u, v),
integral(domain, dot(grad(u), grad(v))) + integral(I, expr_I))
# # TODO BUG
# bnd_A = A.get_boundary(axis=0, ext=1)
#
# a = BilinearForm((u,v), integral(domain, dot(grad(u),grad(v)))
# + integral(I, jump(u) * jump(v))
# + integral(bnd_A, dx(u)*v))
expr = TerminalExpr(a)
print(expr)
def test_interface_integral_3():
# ...
A = Square('A')
B = Square('B')
C = Square('C')
AB = A.join(B,
name='AB',
bnd_minus=A.get_boundary(axis=0, ext=1),
bnd_plus=B.get_boundary(axis=0, ext=-1))
domain = AB.join(C,
name='domain',
bnd_minus=B.get_boundary(axis=0, ext=1),
bnd_plus=C.get_boundary(axis=0, ext=-1))
# ...
x, y = domain.coordinates
V = ScalarFunctionSpace('V', domain, kind=None)
assert (V.is_broken)
u, v = elements_of(V, names='u, v')
# ...
I = domain.interfaces
# print(I)
# print(integral(I, jump(u) * jump(v)))
# a = BilinearForm((u,v), integral(domain, u*v))
# a = BilinearForm((u,v), integral(domain, dot(grad(u),grad(v))))
# a = BilinearForm((u,v), integral(I, jump(u) * jump(v)))
a = BilinearForm((u, v),
integral(domain, dot(grad(u), grad(v))) +
integral(I,
jump(u) * jump(v)))
expr = TerminalExpr(a)
print(expr)
# ...
# ... linear forms
b = LinearForm(
v,
integral(domain,
sin(x + y) * v) + integral(I,
cos(x + y) * jump(v)))
expr = TerminalExpr(b)
print(expr)
def test_bilinear_form_2d_2():
domain = Domain('Omega', dim=2)
B1 = Boundary(r'\Gamma_1', domain)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
V = VectorFunctionSpace('V', domain)
u, u1, u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
# ...
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
a = BilinearForm((u, v), int_0(dot(u, v)))
assert (a.domain == domain.interior)
assert (a(u1, v1) == int_0(dot(u1, v1)))
# ...
# ...
a = BilinearForm((u, v), int_0(dot(u, v) + inner(grad(u), grad(v))))
assert (a.domain == domain.interior)
assert (a(u1, v1) == int_0(dot(u1, v1)) + int_0(inner(grad(u1), grad(v1))))
# ...
# ...
a1 = BilinearForm((u1, v1), int_0(dot(u1, v1)))
a = BilinearForm((u, v), a1(u, v))
assert (a.domain == domain.interior)
assert (a(u2, v2) == int_0(dot(u2, v2)))
# ...
# ...
a1 = BilinearForm((u1, v1), int_0(dot(u1, v1)))
a2 = BilinearForm((u2, v2), int_0(inner(grad(u2), grad(v2))))
a = BilinearForm((u, v), a1(u, v) + kappa * a2(u, v))
assert (a.domain == domain.interior)
assert (a(u,
v) == int_0(dot(u, v)) + int_0(kappa * inner(grad(u), grad(v))))
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_user_function_2d_1():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
# right hand side
f = Function('f')
V = ScalarFunctionSpace('V', domain)
u, v = [element_of(V, name=i) for i in ['u', 'v']]
int_0 = lambda expr: integral(domain, expr)
# ...
expr = dot(grad(u), grad(v)) + f(x, y) * u * v
a = BilinearForm((v, u), int_0(expr))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
expr = f(x, y) * v
l = LinearForm(v, int_0(expr))
print(l)
print(TerminalExpr(l))
print('')
def test_linearize_form_2d_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_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_linear_form_2d_2():
domain = Domain('Omega', dim=2)
B1 = Boundary(r'\Gamma_1', domain)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
V = VectorFunctionSpace('V', domain)
u, u1, u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
# ...
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
g = Matrix((x, y))
l = LinearForm(v, int_0(dot(g, v)))
assert (l.domain == domain.interior)
assert (l(v1) == int_0(dot(g, v1)))
# ...
# ...
g = Matrix((x, y))
l1 = LinearForm(v1, int_0(dot(g, v1)))
l = LinearForm(v, l1(v))
assert (l.domain == domain.interior)
assert (l(u1) == int_0(dot(g, u1)))
# ...
# ...
g1 = Matrix((x, 0))
g2 = Matrix((0, y))
l1 = LinearForm(v1, int_0(dot(v1, g1)))
l2 = LinearForm(v2, int_0(dot(v2, g2)))
l = LinearForm(v, l1(v) + l2(v))
assert (l.domain == domain.interior)
assert (l(u) == int_0(dot(u, g1)) + int_0(dot(u, g2)))
def test_interface_integral_2():
# ...
A = Square('A')
B = Square('B')
domain = A.join(B,
name='domain',
bnd_minus=A.get_boundary(axis=0, ext=1),
bnd_plus=B.get_boundary(axis=0, ext=-1))
# ...
x, y = domain.coordinates
V = ScalarFunctionSpace('V', domain, kind=None)
assert (V.is_broken)
u, u1, u2, u3 = elements_of(V, names='u, u1, u2, u3')
v, v1, v2, v3 = elements_of(V, names='v, v1, v2, v3')
# ...
I = domain.interfaces
a = BilinearForm((u, v), integral(domain, dot(grad(u), grad(v))))
b = BilinearForm((u, v), integral(I, jump(u) * jump(v)))
A = BilinearForm(((u1, u2), (v1, v2)),
a(u1, v1) + a(u2, v2) + b(u1, v1) + b(u2, v2) + b(u1, v2))
B = BilinearForm(
((u1, u2, u3), (v1, v2, v3)),
a(u1, v1) + a(u2, v2) + a(u3, v3) + b(u1, v1) + b(u2, v2) + b(u1, v2))
print(TerminalExpr(A))
print(TerminalExpr(B))
# ...
# ... linear forms
b = LinearForm(v, integral(I, jump(v)))
b = LinearForm((v1, v2), b(v1) + b(v2))
expr = TerminalExpr(b)
print(expr)
def test_terminal_expr_linear_2d_5(boundary=[r'\Gamma_1', r'\Gamma_3']):
# ... abstract model
domain = Square()
V = ScalarFunctionSpace('V', domain)
B_neumann = [domain.get_boundary(i) for i in boundary]
if len(B_neumann) == 1:
B_neumann = B_neumann[0]
else:
B_neumann = Union(*B_neumann)
x, y = domain.coordinates
nn = NormalVector('nn')
F = element_of(V, name='F')
v = element_of(V, name='v')
u = element_of(V, name='u')
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B_neumann, expr)
expr = dot(grad(v), grad(u))
a = BilinearForm((v, u), int_0(expr))
solution = cos(0.5 * pi * x) * cos(0.5 * pi * y)
f = (1. / 2.) * pi**2 * solution
expr = f * v
l0 = LinearForm(v, int_0(expr))
expr = v * dot(grad(solution), nn)
l_B_neumann = LinearForm(v, int_1(expr))
expr = l0(v) + l_B_neumann(v)
l = LinearForm(v, expr)
print(TerminalExpr(l))
print('')
def test_linearize_form_2d_1():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
W = VectorFunctionSpace('W', domain)
v, F, u = elements_of(V, names='v, F, u')
w, G, m = elements_of(W, names='w, G, m')
int_0 = lambda expr: integral(domain, expr)
# ...
l = LinearForm(v, int_0(F**2 * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(2 * F * u * v)
# ...
# ...
l = LinearForm(v, int_0(dot(grad(F), grad(F)) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(2 * dot(grad(F), grad(u)) * v)
# ...
# ...
l = LinearForm(v, int_0(exp(-F) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(-exp(-F) * u * v)
# ...
# ...
l = LinearForm(v, int_0(cos(F) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(-sin(F) * u * v)
# ...
# ...
l = LinearForm(v, int_0(cos(F**2) * v))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(-2 * F * sin(F**2) * u * v)
# ...
# ...
l = LinearForm(v, int_0(F**2 * dot(grad(F), grad(v))))
a = linearize(l, F, trials=u)
assert a(u, v) == int_0(2 * F * u * dot(grad(F), grad(v)) +
F**2 * dot(grad(u), grad(v)))
# ...
# ...
l = LinearForm(w, int_0(dot(rot(G), grad(G)) * w))
a = linearize(l, G, trials=m)
assert a(m, w) == int_0((dot(rot(m), grad(G)) + dot(rot(G), grad(m))) * w)
def test_terminal_expr_linear_2d_3():
domain = Square()
B = domain.boundary
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
nn = NormalVector('nn')
V = ScalarFunctionSpace('V', domain)
u, u1, u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
# ...
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B, expr)
l = LinearForm(v, int_1(dot(grad(v), nn)))
print(TerminalExpr(l))
print('')
def test_tensorize_2d_2():
domain = Domain('Omega', dim=2)
V = VectorFunctionSpace('V', domain)
u, v = elements_of(V, names='u, v')
int_0 = lambda expr: integral(domain , expr)
# ...
# a = BilinearForm((u,v), dot(u,v))
a = BilinearForm((u,v), int_0(curl(u)*curl(v) + div(u)*div(v)))
expr = TensorExpr(a, domain=domain)
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_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_linearize_form_2d_2():
domain = Domain('Omega', dim=2)
V = ScalarFunctionSpace('V', domain)
v, F, u = elements_of(V, names='v, F, u')
int_0 = lambda expr: integral(domain, expr)
# ...
l1 = LinearForm(v, int_0(F**2 * v))
l = LinearForm(v, l1(v))
a = linearize(l, F, trials=u)
expected = linearize(l1, F, trials=u)
assert a == expected
def test_area_2d_1():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
mu = Constant('mu', is_real=True)
e = ElementDomain(domain)
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)
# ...
a = BilinearForm((v, u), int_0(area * u * v))
print(TerminalExpr(a))
def test_tensorize_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, v = elements_of(V, names='u, v')
int_0 = lambda expr: integral(domain, expr)
# ...
# a = BilinearForm((u,v), dot(u,v))
a = BilinearForm((u, v), int_0(curl(u) * curl(v) + div(u) * div(v)))
expr = TensorExpr(a)
print(expr)
def test_tensorize_2d_1():
domain = Domain('Omega', dim=2)
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), int_0(mu*u*v + dot(grad(u),grad(v))))
# a = BilinearForm((u,v), 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_terminal_expr_linear_2d_4():
D1 = InteriorDomain('D1', dim=2)
D2 = InteriorDomain('D2', dim=2)
domain = Union(D1, D2)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
V = ScalarFunctionSpace('V', domain)
u, u1, u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
# ...
int_0 = lambda expr: integral(domain, expr)
l = LinearForm(v, int_0(x * y * v))
print(TerminalExpr(l))
print('')
def test_terminal_expr_bilinear_2d_4():
domain = Domain('Omega', dim=2)
x, y = domain.coordinates
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)
# stokes
a = BilinearForm((u, v), int_0(inner(grad(v), grad(u))))
b = BilinearForm((v, p), int_0(div(v) * p))
a = BilinearForm(((u, p), (v, q)), a(v, u) - b(v, p) + b(u, q))
print(TerminalExpr(a))
print('')
def test_terminal_expr_bilinear_2d_3():
domain = Square()
V = ScalarFunctionSpace('V', domain)
B = domain.boundary
v = element_of(V, name='v')
u = element_of(V, name='u')
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
nn = NormalVector('nn')
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B, expr)
# nitsche
a0 = BilinearForm((u, v), int_0(dot(grad(v), grad(u))))
a_B = BilinearForm((u,v), int_1(-u*dot(grad(v), nn) \
-v*dot(grad(u), nn) \
+kappa*u*v))
a = BilinearForm((u, v), a0(u, v) + a_B(u, v))
print(TerminalExpr(a))
print('')
a = BilinearForm(
(u, v),
int_0(u * v + dot(grad(u), grad(v))) + int_1(v * dot(grad(u), nn)))
print(TerminalExpr(a))
print('')
# ...
# ...
a = BilinearForm((u, v), int_0(u * v))
print(TerminalExpr(a))
print('')
# ...
# ...
a1 = BilinearForm((u, v), int_0(u * v))
a = BilinearForm((u, v), a1(u, v))
print(TerminalExpr(a))
print('')
# ...
# ...
a1 = BilinearForm((u, v), int_0(u * v))
a2 = BilinearForm((u, v), int_0(dot(grad(u), grad(v))))
a = BilinearForm((u, v), a1(u, v) + a2(u, v))
print(TerminalExpr(a))
print('')
# ...
# ...
a1 = BilinearForm((u, v), int_0(u * v))
a2 = BilinearForm((u, v), int_0(dot(grad(u), grad(v))))
a = BilinearForm((u, v), a1(u, v) + kappa * a2(u, v))
print(TerminalExpr(a))
print('')
# ...
# ...
a1 = BilinearForm((u, v), int_0(u * v))
a2 = BilinearForm((u, v), int_0(dot(grad(u), grad(v))))
a3 = BilinearForm((u, v), int_1(v * dot(grad(u), nn)))
a = BilinearForm((u, v), a1(u, v) + kappa * a2(u, v) + mu * a3(u, v))
print(TerminalExpr(a))
print('')
def test_terminal_expr_linear_2d_1():
domain = Domain('Omega', dim=2)
B1 = Boundary(r'\Gamma_1', domain)
x, y = domain.coordinates
kappa = Constant('kappa', is_real=True)
mu = Constant('mu', is_real=True)
nn = NormalVector('nn')
V = ScalarFunctionSpace('V', domain)
u, u1, u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
# ...
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
l = LinearForm(v, int_0(x * y * v))
print(TerminalExpr(l))
print('')
# ...
# ...
l = LinearForm(v, int_0(x * y * v + v))
print(TerminalExpr(l))
print('')
# ...
# ...
g = Matrix((x**2, y**2))
l = LinearForm(v, int_1(v * dot(g, nn)))
print(TerminalExpr(l))
print('')
# ...
# ...
g = Matrix((x**2, y**2))
l = LinearForm(v, int_1(v * dot(g, nn)) + int_0(x * y * v))
print(TerminalExpr(l))
print('')
# ...
# ...
l1 = LinearForm(v1, int_0(x * y * v1))
l = LinearForm(v, l1(v))
print(TerminalExpr(l))
print('')
# ...
# ...
g = Matrix((x, y))
l1 = LinearForm(v1, int_0(x * y * v1))
l2 = LinearForm(v2, int_0(dot(grad(v2), g)))
l = LinearForm(v, l1(v) + l2(v))
print(TerminalExpr(l))
print('')
# ...
# ...
l1 = LinearForm(v1, int_0(x * y * v1))
l2 = LinearForm(v1, int_0(v1))
l = LinearForm(v, l1(v) + kappa * l2(v))
print(TerminalExpr(l))
print('')
# ...
# ...
g = Matrix((x**2, y**2))
l1 = LinearForm(v1, int_0(x * y * v1))
l2 = LinearForm(v1, int_0(v1))
l3 = LinearForm(v, int_1(v * dot(g, nn)))
l = LinearForm(v, l1(v) + kappa * l2(v) + mu * l3(v))
print(TerminalExpr(l))
print('')
def test_terminal_expr_bilinear_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)
nn = NormalVector('nn')
V = ScalarFunctionSpace('V', domain)
u, u1, u2 = [element_of(V, name=i) for i in ['u', 'u1', 'u2']]
v, v1, v2 = [element_of(V, name=i) for i in ['v', 'v1', 'v2']]
# ...
int_0 = lambda expr: integral(domain, expr)
int_1 = lambda expr: integral(B1, expr)
a = BilinearForm((u, v), int_0(u * v))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
a = BilinearForm((u, v), int_0(dot(grad(u), grad(v))))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
a = BilinearForm((u, v), int_0(u * v + dot(grad(u), grad(v))))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
a = BilinearForm(
(u, v),
int_0(u * v + dot(grad(u), grad(v))) + int_1(v * dot(grad(u), nn)))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
a = BilinearForm(((u1, u2), (v1, v2)), int_0(u1 * v1 + u2 * v2))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
a1 = BilinearForm((u1, v1), int_0(u1 * v1))
a = BilinearForm((u, v), a1(u, v))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
a1 = BilinearForm((u1, v1), int_0(u1 * v1))
a2 = BilinearForm((u2, v2), int_0(dot(grad(u2), grad(v2))))
a = BilinearForm((u, v), a1(u, v) + a2(u, v))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
a1 = BilinearForm((u1, v1), int_0(u1 * v1))
a2 = BilinearForm((u2, v2), int_0(dot(grad(u2), grad(v2))))
a = BilinearForm((u, v), a1(u, v) + kappa * a2(u, v))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ...
a1 = BilinearForm((u1, v1), int_0(u1 * v1))
a2 = BilinearForm((u2, v2), int_0(dot(grad(u2), grad(v2))))
a3 = BilinearForm((u, v), int_1(v * dot(grad(u), nn)))
a = BilinearForm((u, v), a1(u, v) + kappa * a2(u, v) + mu * a3(u, v))
print(a)
print(TerminalExpr(a))
print('')
# ...
# ... Poisson with Nitsch method
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))
print(a)
print(TerminalExpr(a))
print('')