def do_z_BVP(self):
a = self.a
nu = self.nu
k = self.k
M = self.M
N = self.N
f_expr = self.f_expr
class DummySympyProblem(SympyProblem, PizzaProblem):
def __init__(self):
self.k = k
super().__init__(f_expr=f_expr)
dummy = DummySympyProblem()
def eval_phi0(th):
fourier_series = 0
for m in range(1, M+1):
fourier_series += self.z1_fourier[m-1] * np.sin(m*nu*(th-a))
return fourier_series + self.eval_gq(self.arc_R, th)
self.polarfd = PolarFD(N, a, nu, self.R0, self.arc_R)
if not self.acheat:
zero = lambda r, th: 0
z = self.polarfd.solve(k,
lambda th: self.eval_v_series(self.R0, th),
eval_phi0,
self.problem.eval_phi1,
self.problem.eval_phi2,
zero, zero, zero, zero
#dummy.eval_f_polar, dummy.eval_d_f_r,
#dummy.eval_d2_f_r, dummy.eval_d2_f_th
)
else:
z = np.zeros((N+1)**2)
# Add back v_asympt
self.v = np.zeros((N+1, N+1), dtype=complex)
for m in range(N+1):
for l in range(N+1):
r = self.polarfd.get_r(m)
th = self.polarfd.get_th(l)
index = self.polarfd.get_index(m, l)
self.v[m, l] = z[index]# + self.eval_v_asympt(r, th)
class ZMethod:
M = 7
R0 = .1
def __init__(self, options):
self.options = options
def run(self):
self.problem = self.options['problem']
self.z1cheat = self.options['z1cheat']
self.acheat = self.options['acheat']
self.boundary = self.problem.boundary
self.N = self.options['N']
self.a = PizzaProblem.a
self.nu = PizzaProblem.nu
self.k = self.problem.k
self.arc_R = self.boundary.R + self.boundary.bet
self.do_algebra()
self.calc_z1_fourier()
self.do_z_BVP()
self.create_v_interp()
self.calc_a_coef()
self.do_u_BVP()
result = {
'v': self.v,
'v_error': self.get_v_error(),
'u': self.u,
'u_error': self.u_error,
'polarfd': self.polarfd,
'pert_solver': self.pert_solver,
}
return result
def do_algebra(self):
g = self.problem.g
v_asympt = self.problem.v_asympt
diff = sympy.diff
pi = sympy.pi
k, a, r, th, x = sympy.symbols('k a r th x')
def apply_helmholtz_op(u):
d_u_r = diff(u, r)
return diff(d_u_r, r) + d_u_r / r + diff(u, th, 2) / r**2 + k**2 * u
f = -apply_helmholtz_op(v_asympt)
f0 = -apply_helmholtz_op(g)
# Polynomial with p(0) = p'(0) = p"(0) = 0,
# p(1) = 1 and p'(1) = p"(1) = 0
p = 10 * x**3 - 15 * x**4 + 6 * x**5
q1 = (r**2 * 1/2 * (th-a)**2 * f0.subs(th, a) *
p.subs(x, (a-th)/(2*pi-a)+1))
q2 = (r**2 * 1/2 * (th-2*np.pi)**2 * f0.subs(th, 2*pi) *
p.subs(x, (th-2*pi)/(2*pi-a)+1))
q = q1 + q2
f1 = f0 - apply_helmholtz_op(q)
subs_dict = {
'k': self.k,
'a': self.a,
'nu': self.nu,
}
initial_func = self.problem.v_series - (g+q)
d_initial_func = diff(initial_func, r)
lambdify_modules = SympyProblem.lambdify_modules
def my_lambdify(expr):
# Assuming function is 0 at r=0 ... this may not
# be necessary
lam = sympy.lambdify((r, th),
expr.subs(subs_dict),
modules=lambdify_modules)
def newfunc(r, th):
if r == 0:
return 0
else:
return lam(r, th)
return newfunc
v_series_lambda = my_lambdify(self.problem.v_series)
R0 = self.R0
v_series_error = [self.problem.eval_v(R0, th) - v_series_lambda(R0, th)
for th in np.arange(self.a, 2*np.pi, .1)]
print('v_series_error:', np.max(np.abs(v_series_error)))
self.f_expr = f
self.eval_v_asympt = my_lambdify(v_asympt)
self.eval_initial_func = my_lambdify(initial_func)
self.eval_d_initial_func = my_lambdify(d_initial_func)
self.eval_v_series = my_lambdify(self.problem.v_series)
eval_g = my_lambdify(g)
self.eval_gq = my_lambdify(g+q)
self.eval_f1 = my_lambdify(f1)
"""r_data = np.linspace(0, .1, 512)
f1_data = [self.eval_f1(r, 2*np.pi) for r in r_data]
f1_data += [self.eval_f1(r, self.a) for r in r_data]
f1_max = np.max(np.abs(f1_data))
print('f1_max:', f1_max)
bc_data1 = [self.problem.eval_phi1(r) - self.eval_v_asympt(r, 2*np.pi)
- eval_g(r, 2*np.pi) for r in r_data]
bc_max1 = np.max(np.abs(bc_data1))
print('bc_max1:', bc_max1)
bc_data2 = [self.problem.eval_phi2(r) - self.eval_v_asympt(r, self.a)
- eval_g(r, self.a) for r in r_data]
bc_max2 = np.max(np.abs(bc_data2))
print('bc_max2:', bc_max2)"""
#th = np.pi
#z_data = [self.problem.eval_v(r, th) - self.eval_v_asympt(r, th)
# for r in r_data]
#plt.plot(r_data, z_data)
#plt.show()
def calc_z1_fourier(self):
expected_z1_fourier = self.get_expected_z1_fourier()
if self.z1cheat:
self.z1_fourier = expected_z1_fourier
elif self.z1_fourier is not None:
# Cached from a previous run
pass
else:
self.z1_fourier = ps.ode.calc_z1_fourier(
self.eval_initial_func,
self.eval_d_initial_func,
self.eval_f1, self.a, self.nu, self.k,
self.R0, self.arc_R, self.M
)
print('ODE error:')
print(np.abs(self.z1_fourier - expected_z1_fourier))
def get_expected_z1_fourier(self):
"""
Compute the expected Fourier coefficients of z1 from
z1 = v - (v_asympt g + q)
For checking the accuracy of the ODE method or
skipping the ODE method during testing
"""
R = self.arc_R
def z1_expected(th):
return self.problem.eval_v(R, th) - self.eval_gq(R, th)
# This should be close to 0
return arc_dst(self.a, z1_expected)[:self.M]
def do_z_BVP(self):
a = self.a
nu = self.nu
k = self.k
M = self.M
N = self.N
f_expr = self.f_expr
class DummySympyProblem(SympyProblem, PizzaProblem):
def __init__(self):
self.k = k
super().__init__(f_expr=f_expr)
dummy = DummySympyProblem()
def eval_phi0(th):
fourier_series = 0
for m in range(1, M+1):
fourier_series += self.z1_fourier[m-1] * np.sin(m*nu*(th-a))
return fourier_series + self.eval_gq(self.arc_R, th)
self.polarfd = PolarFD(N, a, nu, self.R0, self.arc_R)
if not self.acheat:
zero = lambda r, th: 0
z = self.polarfd.solve(k,
lambda th: self.eval_v_series(self.R0, th),
eval_phi0,
self.problem.eval_phi1,
self.problem.eval_phi2,
zero, zero, zero, zero
#dummy.eval_f_polar, dummy.eval_d_f_r,
#dummy.eval_d2_f_r, dummy.eval_d2_f_th
)
else:
z = np.zeros((N+1)**2)
# Add back v_asympt
self.v = np.zeros((N+1, N+1), dtype=complex)
for m in range(N+1):
for l in range(N+1):
r = self.polarfd.get_r(m)
th = self.polarfd.get_th(l)
index = self.polarfd.get_index(m, l)
self.v[m, l] = z[index]# + self.eval_v_asympt(r, th)
def calc_a_coef(self):
a = self.a
nu = self.nu
k = self.k
arc_R = self.arc_R
M = self.M
fft_a_coef = self.get_expected_a_coef()
if self.acheat:
self.a_coef = fft_a_coef
else:
eval_phi0 = self.problem.eval_phi0
def eval_bc0(th):
r = self.boundary.eval_r(th)
return eval_phi0(th) - self.v_interp(r, th)
a_coef, singvals = abcoef.calc_a_coef(self.problem,
self.boundary, eval_bc0, self.M, self.problem.get_m1())
self.a_coef = a_coef
np.set_printoptions(precision=4)
if self.boundary.name == 'arc':
error = np.abs(self.a_coef - fft_a_coef)
print('a_coef error:', error)
print()
elif self.problem.name == 'iz-bessel':
print('a_coef:', scipy.real(a_coef))
print()
def get_expected_a_coef(self):
if self.problem.name == 'iz-bessel':
return np.zeros(self.M)
# Arc only
r = self.problem.R
def eval_bc0(th):
return self.problem.eval_phi0(th) - self.problem.eval_v(r, th)
def eval_diff(th):
return self.problem.eval_v(r, th) - self.v_interp(r, th)
interp_error = arc_dst(self.a, eval_diff)[:self.M]
print('b interp error:')
print(interp_error)
print()
b_coef = arc_dst(self.a, eval_bc0)[:self.M]
return abcoef.b_to_a(b_coef, self.k, self.problem.R, self.nu)
def create_v_interp(self):
"""
Create 6th order accurate interpolating function for v
"""
N = self.N
r_data = []
th_data = []
v_real_data = []
v_imag_data = []
# Don't need to interpolate far away from boundary
if N >= 128:
rmin = (self.boundary.R - self.boundary.bet) * (2/3)
else:
rmin = 0
for m in range(N+1):
r = self.polarfd.get_r(m)
if r < rmin:
continue
r_data.append(r)
for l in range(N+1):
th = self.polarfd.get_th(l)
th_data.append(th)
for m in range(N+1):
r = self.polarfd.get_r(m)
if r < rmin:
continue
for l in range(N+1):
th = self.polarfd.get_th(l)
v_real_data.append(self.v[m, l].real)
v_imag_data.append(self.v[m, l].imag)
real_interp = interp2d(r_data, th_data, v_real_data, kind='quintic')
imag_interp = interp2d(r_data, th_data, v_imag_data, kind='quintic')
def interp(r, th):
return complex(real_interp(r, th), imag_interp(r, th))
self.v_interp = interp
def do_u_BVP(self):
a = self.a
nu = self.nu
k = self.k
M = self.M
eval_v_asympt = self.eval_v_asympt
f_expr = self.f_expr
eval_phi0 = self.problem.eval_phi0
eval_phi1 = self.problem.eval_phi1
eval_phi2 = self.problem.eval_phi2
eval_expected_polar = self.problem.eval_expected_polar
a_coef = self.a_coef
_n_basis_dict = self.problem.n_basis_dict
expected_known = self.problem.name == 'iz-bessel'
class u_BVP(SympyProblem, PizzaProblem):
n_basis_dict = _n_basis_dict
def __init__(self):
self.k = k
self.expected_known = expected_known
super().__init__(f_expr=f_expr)
def eval_bc(self, arg, sid):
r, th = domain_util.arg_to_polar(self.boundary, a, arg, sid)
if sid == 0:
bc = eval_phi0(th) - eval_v_asympt(r, th)
for m in range(1, M+1):
bc -= a_coef[m-1] * jv(m*nu, k*r) * np.sin(m*nu*(th-a))
return bc
elif sid == 1 and r >= 0:
return eval_phi1(r) - eval_v_asympt(r, 2*np.pi)
elif sid == 2 and r >= 0:
return eval_phi2(r) - eval_v_asympt(r, a)
return 0
def eval_expected_polar(self, r, th):
return eval_expected_polar(r, th) - eval_v_asympt(r, th)
my_u_BVP = u_BVP()
my_u_BVP.boundary = self.problem.boundary
options = {}
options.update(self.options)
options['problem'] = my_u_BVP
# Save the solver so we can use its grid
self.pert_solver = ps.ps.PizzaSolver(options)
result = self.pert_solver.run()
self.u = result.u_act
self.u_error = result.error
def get_v_error(self):
N = self.N
errors = []
for m in range(1, N):
r = self.polarfd.get_r(m)
expected = np.zeros(N-1)
for l in range(1, N):
th = self.polarfd.get_th(l)
expected[l-1] = self.problem.eval_v(r, th)
to_dst = expected - self.v[m, 1:N]
fourier = dst(to_dst, type=1)[:self.M] / N
errors.append(np.max(np.abs(fourier)))
return max(errors)
prev_error = None
class Boundary:
pass
boundary = Boundary()
boundary.R = R
boundary.bet = 0
for N in [16, 32, 64, 128]:
my_zmethod = ZMethod({})
my_zmethod.N = N
my_zmethod.boundary = boundary
my_polarfd = PolarFD(N, a, nu, R)
my_zmethod.polarfd = my_polarfd
v = np.zeros([N+1, N+1])
for m in range(N+1):
r = my_polarfd.get_r(m)
for l in range(N+1):
th = my_polarfd.get_th(l)
v[m, l] = eval_v(r, th)
my_zmethod.v = v
v_interp = my_zmethod.create_v_interp()
N2 = 4*N
diff = []
for r in np.linspace(R*.75, R, N2):
for l in range(N+1):
new[m, l] = u[my_polarfd.get_index(m, l)]
return new
"""
Perform the convergence test.
"""
prev_error = None
u2 = None
u1 = None
u0 = None
for N in N_list:
print('---- {0} x {0} ----'.format(N))
my_polarfd = PolarFD()
u = my_polarfd.solve(N, k, a, nu, R1, eval_phi0, eval_phi1, eval_phi2,
eval_f, eval_d_f_r, eval_d2_f_r, eval_d2_f_th)
exp = np.zeros((N+1)**2)
for m in range(N+1):
r = my_polarfd.get_r(m)
for l in range(N+1):
th = my_polarfd.get_th(l)
exp[my_polarfd.get_index(m, l)] = eval_u(r, th)
error = np.max(np.abs(u-exp))
print('Error:', error)
if prev_error is not None: