def compute_resonances(elt=None, K0=None, K1=None, K2=None, neigs=0):
"""
Compute resonances via a generalized linear eigenvalue problem.
Input:
elt
K0
K1
K2
neigs -- Number of poles desired (default: 0 --> compute all)
Output:
l
V
"""
if elt is not None:
(N, nnz) = problem_size(elt)
issparse = (neigs != 0) and (nnz < 0.2 * N ** 2) and (N > 100)
(K0, K1, K2) = form_operators(elt, issparse)
N = len(K0)
# TODO: Adding sparsity
Z = np.zeros((N, N))
I = np.eye(N)
A = np.vstack((np.hstack((K0, Z)), np.hstack((Z, I))))
B = np.vstack((np.hstack((-K1, -K2)), np.hstack((I, Z))))
ll = eig(a=A, b=B, left=False, right=False)
ll_sorted = (np.unique(ll.round(decimals=4)))[np.argsort(np.abs(np.unique(ll.round(decimals=4))))]
ll_sorted = ll_sorted[np.abs(ll_sorted) < 1e308]
return ll_sorted * 1.0j
def checked_resonances2(elt, neigs=0, tol=1e-6):
"""
Compute resonances with two densities in order to check convergence.
If the same answer occurs to within tol, accept the pole as converged.
Inputs:
elt - coarse mesh
neigs - number of poles desired? (default: 0 -> compute all)
tol - absolute estimated error tolerance (default: 1e-6)
Use tol = 0 to return everything
"""
(l, V) = compute_resonances(elt, neigs)
if neigs == 0:
neigs = len(l)
N = problem_size(elt)
dl = np.zeros((neigs,1))
V = V[0:N,:]
for k in range(0,neigs):
dl[k] = errest_resonance(elt, l[k], V[:,k])
if tol > 0:
# Filter eigenvalues
is_good = np.abs(dl) < tol
l = l[is_good]
dl = dl[is_good]
V = V[:,is_good]
def plot_potential(elt):
nelt = len(elt)
(N, _) = problem_size(elt)
N += (len(elt) - 1)
u = np.zeros((N,))
x_tot = np.zeros((N,))
base = 0
for el in elt:
order = el['order']
x = np.cos(np.pi * np.arange(order,-1,-1) / order)
xelt = el['a'] * (1-x)/2 + el['b'] * (1+x)/2
x_tot[base:base+order+1] = xelt[:];
u[base:base+order+1] = eval_potential(el, xelt)
base = base+order+1
return (x_tot, u)
def plot_fields(elt, u):
(N, _) = problem_size(elt)
x_tot = None
if len(u) > N:
x_tot = np.zeros((N+len(elt)-1,))
else:
x_tot = np.zeros((N,))
base = 0
for el in elt:
order = el['order']
x = np.cos(np.pi * np.arange(order,-1,-1) / order)
xelt = el['a'] * (1-x)/2 + el['b'] * (1+x)/2
x_tot[base:base+order+1] = xelt
if len(u) > N:
base = base+order+1
else:
base = base+order
return x_tot
def plane_forcing (elt, l):
"""
Compute a forcing vector corresponding to the influence of an
incident wave of the form exp(l*x)
"""
z = l/1.0j
(N, _) = problem_size(elt)
F = np.zeros((N,), dtype='complex_')
base = 0
for el in elt:
order = el['order']
x = np.cos(np.pi * np.arange(order,-1,-1) / order)
xelt = el['a'] * (1-x)/2.0 + el['b'] * (1+x)/2.0
F[base:base+order+1] = -eval_potential(el, xelt) * np.exp(z*xelt)
F[base] = 0
F[base+order] = 0
base = base+order
return F