def interp(self, y, eigvals, eigvectors, eigval=1, verbose=False):
"""Interpolate solution eigenvector and it's derivative onto y
Parameters
----------
y : array
Interpolation points
eigvals : array
All computed eigenvalues
eigvectors : array
All computed eigenvectors
eigval : int, optional
The chosen eigenvalue, ranked with descending imaginary
part. The largest imaginary part is 1, the second
largest is 2, etc.
verbose : bool, optional
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 not len(self.P4) == len(y):
SB = Basis(N, 'C', bc='Biharmonic', quad=self.quad)
self.P4 = SB.evaluate_basis_all(x=y)
self.T4x = SB.evaluate_basis_derivative_all(x=y, k=1)
phi = np.dot(self.P4, phi_hat)
dphidy = np.dot(self.T4x, phi_hat)
return eigval, phi, dphidy
def interp(self, y, eigvals, eigvectors, eigval=1, verbose=False):
"""Interpolate solution eigenvector and it's derivative onto y
Parameters
----------
y : array
Interpolation points
eigvals : array
All computed eigenvalues
eigvectors : array
All computed eigenvectors
eigval : int, optional
The chosen eigenvalue, ranked with descending imaginary
part. The largest imaginary part is 1, the second
largest is 2, etc.
verbose : bool, optional
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 not len(self.P4) == len(y):
SB = Basis(N, 'C', bc='Biharmonic', quad=self.quad)
self.P4 = SB.evaluate_basis_all(x=y)
self.T4x = SB.evaluate_basis_derivative_all(x=y, k=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 = Basis(N, 'C', bc='Biharmonic', quad=self.quad)
SB.plan((N, N), 0, np.float, {})
x, _ = self.x, self.w = SB.points_and_weights(N)
# Trial function
P4 = SB.evaluate_basis_all(x=x)
# Second derivatives
T2x = SB.evaluate_basis_derivative_all(x=x, k=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
def assemble(self):
N = self.N
SB = Basis(N, 'C', bc='Biharmonic', quad=self.quad)
SB.plan((N, N), 0, np.float, {})
x, _ = self.x, self.w = SB.points_and_weights(N)
# Trial function
P4 = SB.evaluate_basis_all(x=x)
# Second derivatives
T2x = SB.evaluate_basis_derivative_all(x=x, k=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
