def eval_expected(self, x, y, **kwargs):
r, th = cart_to_polar(x, y)
if th < self.a/2:
th += 2*np.pi
return self.eval_expected_polar(r, th, **kwargs)
def c1_test(self, plot=True):
"""
Plot the reconstructed Neumann data, as approximated by a Chebyshev
expansion. If the Chebyshev expansion shows "spikes" near the
interfaces of the segments, something is probably wrong.
"""
sample = self.get_boundary_sample()
do_exact = hasattr(self.problem, 'eval_d_u_outwards')
s_data = np.zeros(len(sample))
expansion_data = np.zeros(len(sample))
if do_exact:
exact_data = np.zeros(len(sample))
for l in range(len(sample)):
p = sample[l]
s_data[l] = p['s']
if do_exact:
exact_data[l] = self.problem.eval_d_u_outwards(p['arg'], p['sid']).real
r, th = domain_util.cart_to_polar(p['x'], p['y'])
sid = domain_util.get_sid(self.a, th)
if sid == 0:
arg = th
else:
arg = r
for JJ in range(len(self.B_desc)):
expansion_data[l] +=\
(self.c1[JJ] *
self.eval_dn_B_arg(0, JJ, arg, sid)).real
import matplotlib.pyplot as plt
if do_exact:
error = np.max(np.abs(exact_data - expansion_data))
print('c1 error: {}'.format(error))
plt.plot(s_data, exact_data, linewidth=5, color='#BBBBBB', label='Exact')
if not plot:
return
plt.plot(s_data, expansion_data, label='Expansion')
plt.legend(loc=2)
#plt.ylim(-1.5, 1.5)
plt.title('c1')
plt.xlabel('Arclength s')
plt.ylabel('Reconstructed Neumann data')
plt.show()
def do_extend_2_standard(self, i, j, **kwargs):
x, y = self.get_coord(i, j)
x0, y0 = self.get_radius_point(2, x, y)
param_r = cart_to_polar(x0, y0)[0]
deriv_types = (
(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (4, 0),
)
derivs = self.ext_calc_xi_derivs(deriv_types, param_r, 2, **kwargs)
n = self.signed_dist_to_radius(2, x, y)
value = self.extend_radius(n=n, **derivs)
return {'elen': abs(n), 'value': value}
def _calc_inhomo_radius(self, x0, y0, x1, y1, tan, n):
p = self.problem
f = p.eval_f(x0, y0)
if x0 == 0 and y0 == 0:
"""
Use Taylor's theorem to construct a smooth extension of
grad_f, and hessian_f at the origin, since they may
be undefined or annoying to calculate at this point.
"""
h = self.AD_len / (10 * self.N)
hessian_f = p.eval_hessian_f(-h, 0)
grad_f = p.eval_grad_f(-h, 0)
grad_f += h * hessian_f.dot((1, 0))
else:
grad_f = p.eval_grad_f(x0, y0)
hessian_f = p.eval_hessian_f(x0, y0)
vec = np.array((x1 - x0, y1 - y0))
if np.linalg.norm(vec) != 0:
vec /= np.linalg.norm(vec)
d_f_n = grad_f.dot(vec) * np.sign(n)
d2_f_n = hessian_f.dot(vec).dot(vec)
d2_f_s = hessian_f.dot(tan).dot(tan)
elen = abs(n)
# Only so that elen matches that of the homogeneous extension
r, th = cart_to_polar(x0, y0)
if x0 < 0 or r > self.R:
elen += r
v = self.inhomo_extend_arbitrary(n=n, f=f, d_f_n=d_f_n, d2_f_n=d2_f_n, d2_f_s=d2_f_s, curv=0)
return {"elen": elen, "value": v}
def c0_test(self, plot=True):
"""
Plot the Dirichlet data along with its Chebyshev expansion. If there
is more than a minor difference between the two, something is wrong.
"""
k = self.k
a = self.a
nu = self.nu
sample = self.get_boundary_sample()
s_data = np.zeros(len(sample))
exact_data = np.zeros(len(sample))
expansion_data = np.zeros(len(sample))
exact_data_outer = []
expansion_data_outer = []
for l in range(len(sample)):
p = sample[l]
s_data[l] = p['s']
r, th = domain_util.cart_to_polar(p['x'], p['y'])
sid = domain_util.get_sid(self.a, th)
exact_data[l] = self.problem.eval_bc(p['arg'], sid).real
if sid == 0:
arg = th
else:
arg = r
for JJ in range(len(self.B_desc)):
expansion_data[l] +=\
(self.c0[JJ] *
self.eval_dn_B_arg(0, JJ, arg, sid)).real
if sid == 0:
exact_data_outer.append(exact_data[l])
expansion_data_outer.append(expansion_data[l])
exact_data_outer = np.array(exact_data_outer)
expansion_data_outer = np.array(expansion_data_outer)
print('c0 error outer:', np.max(np.abs(exact_data_outer - expansion_data_outer)))
print('c0 error:', np.max(np.abs(exact_data - expansion_data)))
#for l in range(len(exact_data)):
# diff = abs(exact_data[l] - expansion_data[l])
# if diff > .06:
# print(sample[l], 'diff:', diff)
if not plot:
return
import matplotlib.pyplot as plt
plt.plot(s_data, exact_data, linewidth=5, color='#BBBBBB', label='Exact')
plt.plot(s_data, expansion_data, label='Expansion')
plt.legend(loc=0)
plt.title('c0')
plt.xlabel('Arclength s')
plt.ylabel('Dirichlet data')
plt.show()
def eval_f(self, x, y):
if self.homogeneous:
return 0
return self.eval_f_polar(*cart_to_polar(x, y))
def get_polar(self, i, j):
"""
Get the polar coordinates of grid point (i,j)
"""
x, y = self.get_coord(i, j)
return cart_to_polar(x, y)
def eval_grad_f(self, x, y):
"""
Evaluate gradient of f at (x, y), using the chain rule. Returns the
result as a NumPy array.
"""
return self.eval_grad_f_polar(*cart_to_polar(x, y))
def eval_hessian_f(self, x, y):
"""
Evaluate the Cartesian Hessian of f at (x, y), using the chain rule.
Returns the result as a 2x2 NumPy array.
"""
return self.eval_hessian_f_polar(*cart_to_polar(x, y))
