def u_glob(U, cells, vertices, dof_map, resolution_per_element=51):
"""
Compute (x, y) coordinates of a curve y = u(x), where u is a
finite element function: u(x) = sum_i of U_i*phi_i(x).
(The solution of the linear system is in U.)
Method: Run through each element and compute curve coordinates
over the element.
This function works with cells, vertices, and dof_map.
"""
x_patches = []
u_patches = []
nodes = {} # node coordinates (use dict to avoid multiple values)
for e in range(len(cells)):
Omega_e = (vertices[cells[e][0]], vertices[cells[e][-1]])
d = len(dof_map[e]) - 1
phi = basis(d)
X = np.linspace(-1, 1, resolution_per_element)
x = affine_mapping(X, Omega_e)
x_patches.append(x)
u_cell = 0 # u(x) over this cell
for r in range(d+1):
i = dof_map[e][r] # global dof number
u_cell += U[i]*phi[r](X)
u_patches.append(u_cell)
# Compute global coordinates of local nodes,
# assuming all dofs corresponds to values at nodes
X = np.linspace(-1, 1, d+1)
x = affine_mapping(X, Omega_e)
for r in range(d+1):
nodes[dof_map[e][r]] = x[r]
nodes = np.array([nodes[i] for i in sorted(nodes)])
x = np.concatenate(x_patches)
u = np.concatenate(u_patches)
return x, u, nodes
def u_glob(U, vertices, cells, dof_map,
resolution_per_element=51):
"""
Compute (x, y) coordinates of a curve y = u(x), where u is a
finite element function: u(x) = sum_i of U_i*phi_i(x).
Method: Run through each element and compute curve coordinates
over the element.
"""
x_patches = []
u_patches = []
for e in range(len(cells)):
Omega_e = [vertices[cells[e][0]], vertices[cells[e][1]]]
local_nodes = dof_map[e]
d = len(local_nodes) - 1
X = np.linspace(-1, 1, resolution_per_element)
x = affine_mapping(X, Omega_e)
x_patches.append(x)
u_element = 0
for r in range(len(local_nodes)):
i = local_nodes[r] # global node number
u_element += U[i]*phi_r(r, X, d)
u_patches.append(u_element)
x = np.concatenate(x_patches)
u = np.concatenate(u_patches)
return x, u