def test_funspace_2():
'''Test of FunctionSpace.int_phi_phi made up of linear elements, basic
integration
'''
ox, dx, nx = 0., 1., 1
mesh = Mesh(type='uniform', lx=dx, nx=nx, block='B1')
V = FunctionSpace(mesh, {'B1': Element(type='link2')})
# basic properties
assert V.num_dof == (nx + 1)
assert V.size == nx
X = np.linspace(ox, dx, nx+1)
assert allclose(V.X, X)
# Integrate[N(x) N(x) {x, 0, 1}]
fun = lambda x: x
f = lambda i, j: integrate(fun(x) * N[i] * N[j], (x, ox, dx))
ans = V.int_phi_phi(fun=fun, derivative=(False, False))
exact = Matrix(2, 2, lambda i, j: f(i,j))
assert areclose(exact, ans)
# Trivial check with coefficient
fun = lambda x: 1.
a1 = V.int_phi_phi()
a2 = V.int_phi_phi(fun=fun)
assert areclose(a1, a2)
def test_funspace_8():
'''Testing of FunctionSpace.int_phi_phi, MMS'''
ox, dx, nx = 0., 10., 10
mesh = Mesh(type='uniform', lx=dx, nx=nx, block='B1')
V = FunctionSpace(mesh, {'B1': Element(type='link2')})
rhs = np.random.rand(V.num_dof)
A = V.int_phi_phi()
f = lambda x: np.interp(x, V.X, rhs)
b = V.int_phi(f)
assert areclose(np.dot(A, rhs), b)
return
def test_funspace_7():
'''Test of FunctionSpace.int_phi_phi made up of linear elements, mixed
matrix integration
'''
ox, dx, nx = 0., 1., 1
mesh = Mesh(type='uniform', lx=dx, nx=nx, block='B1')
V = FunctionSpace(mesh, {'B1': Element(type='link2')})
# Integrate[N(x) N'(x) {x, 0, 1}]
f = lambda i, j: integrate(N[i] * dN[j], (x, ox, dx))
ans = V.int_phi_phi(derivative=(False, True))
exact = Matrix(2, 2, lambda i, j: f(i,j))
assert areclose(exact, ans)
# Integrate[N'(x) N(x) {x, 0, 1}]
f = lambda i, j: integrate(dN[i] * N[j], (x, ox, dx))
ans = V.int_phi_phi(derivative=(True, False))
exact = Matrix(2, 2, lambda i, j: f(i,j))
assert areclose(exact, ans)
return
def test_funspace_4():
'''Test of FunctionSpace.int_phi_phi made up of linear elements, Laplace
matrix multiply
'''
ox, dx, nx = 0., 1., 10
mesh = Mesh(type='uniform', lx=dx, nx=nx, block='B1')
V = FunctionSpace(mesh, {'B1': Element(type='link2')})
C = V.int_phi_phi(derivative=(True, True))
sol = np.ones(V.num_dof)
b = np.dot(C, sol)
assert norm(b) < 1.e-12
def test_funspace_6():
'''Test of FunctionSpace.int_phi_phi made up of linear elements, Laplace
norm check. Solution = x^2
'''
ox, dx, nx = 0., 1., 10
mesh = Mesh(type='uniform', lx=dx, nx=nx, block='B1')
V = FunctionSpace(mesh, {'B1': Element(type='link2')})
C = V.int_phi_phi(derivative=(True, True))
sol = V.X ** 2
b = np.dot(C, sol)
rhs = V.int_phi(lambda x: 2.)
# natural b.c. not satisfied on right, don't check it
rhs[-1] = -b[-1]
assert norm(rhs + b) < 1.e-12
def test_funspace_9():
'''Testing of FunctionSpace.int_phi_phi, mixed integration MMS'''
ox, dx, nx = 0., 10., 10
mesh = Mesh(type='uniform', lx=dx, nx=nx, block='B1')
V = FunctionSpace(mesh, {'B1': Element(type='link2')})
D = V.int_phi_phi(derivative=(False,True))
sol = np.ones(V.num_dof)
b = np.dot(D, sol)
# Norm check (rhs d/dx + Neumann, const soln)
assert norm(b) < 1.e-12
D[0, 0] = 1.0
D[0, 1:] = 0.0
D[-1, -1] = 1.0
D[-1, 0:-1] = 0.0
sol = V.X
b = np.dot(D, sol)
rhs = V.int_phi(lambda x: 1)
rhs[0] = sol[0]
rhs[-1] = sol[-1]
# norm check (d/dx+Dirichlet sol=x)
assert norm(rhs - b) < 1.e-12
