def interp(self, y, eigval=1, same_mesh=False, verbose=False):
"""Interpolate solution eigenvector and it's derivative onto y
"""
N = self.N
nx, eigval = self.get_eigval(eigval, verbose)
phi_hat = np.zeros(N, np.complex)
phi_hat[:-4] = np.squeeze(self.eigvectors[:, nx])
if same_mesh:
phi = np.zeros_like(phi_hat)
dphidy = np.zeros_like(phi_hat)
if self.SB is None:
self.SB = ShenBiharmonicBasis(N, quad=self.quad, plan=True)
self.SD = ShenDirichletBasis(N, quad=self.quad, plan=True)
self.CDB = inner_product((self.SD, 0), (self.SB, 1))
phi = self.SB.ifst(phi_hat, phi)
dphidy_hat = self.CDB.matvec(phi_hat)
dphidy_hat = self.SD.apply_inverse_mass(dphidy_hat)
dphidy = self.SD.backward(dphidy_hat, dphidy)
else:
# Recompute interpolation matrices if necessary
if not len(self.P4) == len(y):
SB = ShenBiharmonicBasis(N, quad=self.quad)
V = self.V = SB.vandermonde(y)
P4 = self.P4 = SB.get_vandermonde_basis(V)
T4x = self.T4x = SB.get_vandermonde_basis_derivative(V, 1)
phi = np.dot(self.P4, phi_hat)
dphidy = np.dot(self.T4x, phi_hat)
return phi, dphidy
def interp(self,
y,
eigvals,
eigvectors,
eigval=1,
same_mesh=False,
verbose=False):
"""Interpolate solution eigenvector and it's derivative onto y
args:
y Interpolation points
eigvals All computed eigenvalues
eigvectors All computed eigenvectors
kwargs:
eigval The chosen eigenvalue, ranked with descending imaginary
part. The largest imaginary part is 1, the second
largest is 2, etc.
same_mesh Boolean. Whether or not to interpolate to the same
quadrature points as used for computing the eigenvectors
verbose Boolean. Print information or not
"""
N = self.N
nx, eigval = self.get_eigval(eigval, eigvals, verbose)
phi_hat = np.zeros(N, np.complex)
phi_hat[:-4] = np.squeeze(eigvectors[:, nx])
if same_mesh:
phi = np.zeros_like(phi_hat)
dphidy = np.zeros_like(phi_hat)
if self.SB is None:
self.SB = ShenBiharmonicBasis(N, quad=self.quad, plan=True)
self.SD = ShenDirichletBasis(N, quad=self.quad, plan=True)
self.CDB = inner_product((self.SD, 0), (self.SB, 1))
phi = self.SB.ifst(phi_hat, phi)
dphidy_hat = self.CDB.matvec(phi_hat)
dphidy_hat = self.SD.apply_inverse_mass(dphidy_hat)
dphidy = self.SD.backward(dphidy_hat, dphidy)
else:
# Recompute interpolation matrices if necessary
if not len(self.P4) == len(y):
SB = ShenBiharmonicBasis(N, quad=self.quad)
V = SB.vandermonde(y)
self.P4 = SB.get_vandermonde_basis(V)
self.T4x = SB.get_vandermonde_basis_derivative(V, 1)
phi = np.dot(self.P4, phi_hat)
dphidy = np.dot(self.T4x, phi_hat)
return eigval, phi, dphidy
def assemble(self):
N = self.N
SB = ShenBiharmonicBasis(N, quad=self.quad)
SB.plan((N, N), 0, np.float, {})
x, w = self.x, self.w = SB.points_and_weights(N)
V = SB.vandermonde(x)
# Trial function
P4 = SB.get_vandermonde_basis(V)
# Second derivatives
T2x = SB.get_vandermonde_basis_derivative(V, 2)
# (u'', v)
K = np.zeros((N, N))
K[:-4, :-4] = inner_product((SB, 0), (SB, 2)).diags().toarray()
# ((1-x**2)u, v)
xx = np.broadcast_to((1 - x**2)[:, np.newaxis], (N, N))
#K1 = np.dot(w*P4.T, xx*P4) # Alternative: K1 = np.dot(w*P4.T, ((1-x**2)*P4.T).T)
K1 = np.zeros((N, N))
K1 = SB.scalar_product(xx * P4, K1)
K1 = extract_diagonal_matrix(
K1).diags().toarray() # For improved roundoff
# ((1-x**2)u'', v)
K2 = np.zeros((N, N))
K2 = SB.scalar_product(xx * T2x, K2)
K2 = extract_diagonal_matrix(
K2).diags().toarray() # For improved roundoff
# (u'''', v)
Q = np.zeros((self.N, self.N))
Q[:-4, :-4] = inner_product((SB, 0), (SB, 4)).diags().toarray()
# (u, v)
M = np.zeros((self.N, self.N))
M[:-4, :-4] = inner_product((SB, 0), (SB, 0)).diags().toarray()
Re = self.Re
a = self.alfa
B = -Re * a * 1j * (K - a**2 * M)
A = Q - 2 * a**2 * K + a**4 * M - 2 * a * Re * 1j * M - 1j * a * Re * (
K2 - a**2 * K1)
return A, B