def setUp(self):
"""Construct a square mesh and it's cotangent Laplacian/Area matrix
Sparse representations are at self.A and self.L_c, dense versions
for matrix comparision are at self.A_d. self.L_c_d
"""
c = np.array([[ 0., 0., 0.],
[ 1., 0., 0.],
[ 0., 1., 0.],
[-1., 0., 0.],
[ 0.,-1., 0.]])
cI = np.array([[0, 4, 1],
[3, 0, 2],
[1, 2, 0],
[0, 3, 4]], dtype=np.uint32)
self.square_mesh = Face(coords=c,coordsIndex=cI)
self.L_c, self.A = self.square_mesh.laplacian_operator()
self.L_c_d = self.L_c.todense()
self.A_d = self.A.todense()
class LaplacianTest(unittest.TestCase):
"""Unit tests for Laplacian Operator"""
def setUp(self):
"""Construct a square mesh and it's cotangent Laplacian/Area matrix
Sparse representations are at self.A and self.L_c, dense versions
for matrix comparision are at self.A_d. self.L_c_d
"""
c = np.array([[ 0., 0., 0.],
[ 1., 0., 0.],
[ 0., 1., 0.],
[-1., 0., 0.],
[ 0.,-1., 0.]])
cI = np.array([[0, 4, 1],
[3, 0, 2],
[1, 2, 0],
[0, 3, 4]], dtype=np.uint32)
self.square_mesh = Face(coords=c,coordsIndex=cI)
self.L_c, self.A = self.square_mesh.laplacian_operator()
self.L_c_d = self.L_c.todense()
self.A_d = self.A.todense()
def test_cotangent_flat_square_laplacian(self):
L_c_analytic = np.array([[ 4., -1., -1., -1., -1.],
[-1., 1., 0., 0., 0.],
[-1., 0., 1., 0., 0.],
[-1., 0., 0., 1., 0.],
[-1., 0., 0., 0., 1.]])
self.assertTrue(matrix_equal(self.L_c_d, L_c_analytic))
def test_cotangent_flat_square_area(self):
A_analytic = np.diag([2., 1., 1., 1., 1.])/3.
self.assertTrue(matrix_equal(self.A_d, A_analytic))
def test_cotangent_flat_square_euler_step(self):
u_0 = np.zeros(self.square_mesh.n_coords)
u_0[0] = 1
L_c = -1.0*self.L_c
t = self.square_mesh.mean_edge_length
u_t = linalg.spsolve(self.A - t*(self.L_c), u_0)
# heat should spread equally to each corner
self.assertTrue(np.var(u_t[1:]) < 0.000001)
# there should be some heat everywhere
self.assertTrue(np.all(u_t > 0))
def test_cotangent_flat_square_heat_divergence(self):
u_0 = np.zeros(self.square_mesh.n_coords)
u_0[0] = 1
L_c = -1.0*self.L_c
t = self.square_mesh.mean_edge_length
u_t = linalg.spsolve(self.A - t*(self.L_c), u_0)
grad_u_t = self.square_mesh.gradient(u_t)
print grad_u_t
grad_u_t_analytic = np.array([[ 0.15206873, -0.15206873, 0.],
[-0.15206873, 0.15206873, 0.],
[ 0.15206873, 0.15206873, 0.],
[-0.15206873, -0.15206873, 0.]])
self.assertTrue(matrix_equal(grad_u_t, grad_u_t_analytic))
def test_flat_square_mean_length(self):
t = self.square_mesh.mean_edge_length
self.assertTrue(np.absolute(t - (1 + np.sqrt(2))/2.0) < 0.00000000001)