def facet_length(facet):
""" calculate FEniCS facet length """
v_idx = facet.entities(0)
v0 = Vertex(facet.mesh(), v_idx[0])
v1 = Vertex(facet.mesh(), v_idx[1])
x0 = v0.point().x()
x1 = v1.point().x()
y0 = v0.point().y()
y1 = v1.point().y()
return ((x1 - x0)**2 + (y1 - y0)**2)**0.5
def hat_function_grad(vertex, cell):
"""Compute L^\infty-norm of gradient of hat function on 'cell'
and value 1 in 'vertex'."""
# TODO: fix using ghosted mesh
not_working_in_parallel("function 'hat_function_grad'")
assert vertex in vertices(cell), "vertex not in cell!"
# Find adjacent facet
f = [f for f in facets(cell) if not vertex in vertices(f)]
assert len(f) == 1, "Something strange with adjacent cell!"
f = f[0]
# Get unit normal
n = f.normal()
n /= n.norm()
# Pick some vertex on facet
# FIXME: Is it correct index in parallel?
facet_vertex_0 = Vertex(cell.mesh(), f.entities(0)[0])
# Compute signed distance from vertex to facet plane
d = (facet_vertex_0.point() - vertex.point()).dot(n)
# Return norm of gradient
assert d != 0.0, "Degenerate cell!"
return 1.0/abs(d)
def myassemble(mesh):
# Define basis functions and their gradients on the reference elements.
def phi0(x):
return 1.0 - x[0] - x[1]
def phi1(x):
return x[0]
def phi2(x):
return x[1]
def f(x):
return 1.0
phi = [phi0, phi1, phi2]
dphi = np.array(([-1.0, -1.0], [1.0, 0.0], [0.0, 1.0]))
# Define quadrature points
midpoints = np.array(([0.5, 0.0], [0.5, 0.5], [0.0, 0.5]))
N = mesh.num_vertices()
A = np.zeros((N, N))
b = np.zeros((N, 1))
# Used to hold cell vertices
coord = np.zeros([3, 2])
# Iterate over all cells, adding integral contribution to matrix/vector
for c in cells(mesh):
# Extract node numbers and vertex coordinates
nodes = c.entities(0).astype('int')
for i in range(0, 3):
v = Vertex(mesh, int(nodes[i]))
for j in range(0, 2):
coord[i][j] = v.point()[j]
# Compute Jacobian of map and area of cell
J = np.outer(coord[0, :], dphi[0]) + \
np.outer(coord[1, :], dphi[1]) + \
np.outer(coord[2, :], dphi[2])
dx = 0.5 * abs(np.linalg.det(J))
# Iterate over quadrature points
for p in midpoints:
# Map p to physical cell
x = coord[0, :] * phi[0](p) + \
coord[1, :] * phi[1](p) + \
coord[2, :] * phi[2](p)
# Iterate over test functions
for i in range(0, 3):
v = phi[i](p)
dv = np.linalg.solve(J.transpose(), dphi[i])
# Assemble vector (linear form)
integral = f(x)*v*dx / 3.0
b[nodes[i]] += integral
# Iterate over trial functions
for j in range(0, 3):
u = phi[j](p)
du = np.linalg.solve(J.transpose(), dphi[j])
# Assemble matrix (bilinear form)
integral = (np.inner(du, dv)) * dx / 3.0
integral += u * v * dx / 3.0
A[nodes[i], nodes[j]] += integral
return A, b
def edge_to_vector(edge):
v0, v1 = edge.entities(0)
mesh = edge.mesh()
v0 = Vertex(mesh, v0)
v1 = Vertex(mesh, v1)
return point_to_array(v1.point() - v0.point())