def j0zero(s):
""" method that computes root number s, s = 1,2,...,
of the Besselfunction J0 where zj0 is the root,
using table-values, asymptotic formulae and Newton-Raphsons method.
Rellative error app. 1.0e-14 to 1.0e-15
Args:
s(integer): solutian array from last iteration
Returns:
zj0(float): root No. s of besselfunction
"""
sz = np.array([2.4048225557695773, 5.520078110286311, 8.653727912911012, 11.79153443901428, 14.93091770848779])
b0 = (s - 0.25)*np.pi
b08 = 0.125/b0
b082 = b08**2
z0 = b0 + b08*(1. - b082*(124./3 - b082*120928./15))
if s <= 5:
zj0 = sz[s - 1]
elif s > 30:
zj0 = z0
else:
dz0 = besselj(0, z0)/besselj(1, z0)
z0 = z0 + dz0
zj0 = z0
return zj0
def j0zero(s):
""" method that computes root number s, s = 1,2,...,
of the Besselfunction J0 where zj0 is the root,
using table-values, asymptotic formulae and Newton-Raphsons method.
Rellative error app. 1.0e-14 to 1.0e-15
Args:
s(integer): solutian array from last iteration
Returns:
zj0(float): root No. s of besselfunction
"""
sz = np.array([
2.4048225557695773, 5.520078110286311, 8.653727912911012,
11.79153443901428, 14.93091770848779
])
b0 = (s - 0.25) * np.pi
b08 = 0.125 / b0
b082 = b08**2
z0 = b0 + b08 * (1. - b082 * (124. / 3 - b082 * 120928. / 15))
if s <= 5:
zj0 = sz[s - 1]
elif s > 30:
zj0 = z0
else:
dz0 = besselj(0, z0) / besselj(1, z0)
z0 = z0 + dz0
zj0 = z0
return zj0
def hankel_transform(wavefield, max_radius, bessel_zeros, n_zeros=0, multipool=True):
n = len(wavefield)
if n_zeros <= 0: n_zeros = n
wavefield = wavefield.conjugate().T
m1 = (numpy.abs(besselj(1, bessel_zeros[:n])) / max_radius).conjugate().T
max_frequency = bessel_zeros[n] / (2 * numpy.pi * max_radius)
m2 = m1 * max_radius / max_frequency
f = numpy.divide(wavefield, m1)
bessel_zeros_nz = bessel_zeros[0:n_zeros]
bessel_zeros_n = bessel_zeros[n]
f_nz = f[0:n_zeros]
bessel_jn = numpy.abs(besselj(1, bessel_zeros_nz)) / (2 / bessel_zeros_n)
bessel_jm = numpy.abs(besselj(1, bessel_zeros))
if multipool:
args = __create_args(__N_CPU, bessel_zeros[0:n], bessel_jm[0:n], bessel_zeros_nz, bessel_zeros_n, bessel_jn, f_nz, n)
p = multiprocessing.Pool(__N_CPU)
result = numpy.concatenate(p.map(__calculate, args))
p.close()
else:
result = __v_inner_loop(bessel_zeros[0:n], bessel_jm[0:n], bessel_zeros_nz, bessel_zeros_n, bessel_jn, f_nz)
result = result.conjugate().T
result = numpy.multiply(result, m2)
result = result.conjugate().T
return result
def _compute_kernel_2d(R, shape):
J = np.mgrid[[slice(-(s//2), (s+1)//2) for s in shape]]
piabsJ = np.pi * np.linalg.norm(J, axis=0)
Jzero = tuple(s//2 for s in shape)
K_hat = (2*R)**(-1) * R**2 / (piabsJ**2 - R**2) * (
1 + 1j*np.pi/2 * (
piabsJ * besselj(1, piabsJ) * hankel1(0, R) -
R * besselj(0, piabsJ) * hankel1(1, R)
)
)
K_hat[Jzero] = -1/(2*R) + 1j*np.pi/4 * hankel1(1, R)
K_hat[piabsJ == R] = 1j*np.pi*R/8 * (
besselj(0, R) * hankel1(0, R) + besselj(1, R) * hankel1(1, R)
)
return 2 * R * fftshift(K_hat)
def HankelAccept(wavelength, magfield, thickness, q, Iq, theta, modelname):
#==============================================================================
# HankelTransform with fixed circular acceptance angle (circular aperture) for Time of Flight SESANS
#==============================================================================
#acceptq is the q-space needed to create limited acceptance effect
SElength= wavelength*magfield
G = np.zeros_like(SElength, 'd')
threshold=2*pi*theta/wavelength
for i, SElength_i in enumerate(SElength):
allq=np.linspace() #for calculating the Q-range of the scattering power integral
allIq=np.linspace() # This is the model applied to the allq q-space. Needs to refference the model somehow
alldq = (allq[1]-allq[0])*1e10
sigma[i]=wavelength[i]^2*thickness/2/pi*np.sum(allIq*allq*alldq)
s[i]=1-exp(-sigma)
dq = (q[1]-q[0])*1e10
a = (x<threshold)
acceptq = a*q
acceptIq = a*Iq
G[i] = np.sum(besselj(0, acceptq*SElength_i)*acceptIq*acceptq*dq)
# G[i]=np.sum(integral)
G *= dq*1e10*2*pi
P = exp(thickness*wavelength**2/(4*pi**2)*(G-G[0]))
def HankelAccept(wavelength, magfield, thickness, q, Iq, theta, modelname):
#==============================================================================
# HankelTransform with fixed circular acceptance angle (circular aperture) for Time of Flight SESANS
#==============================================================================
#acceptq is the q-space needed to create limited acceptance effect
SElength = wavelength * magfield
G = np.zeros_like(SElength, 'd')
threshold = 2 * pi * theta / wavelength
for i, SElength_i in enumerate(SElength):
allq = np.linspace(
) #for calculating the Q-range of the scattering power integral
allIq = np.linspace(
) # This is the model applied to the allq q-space. Needs to refference the model somehow
alldq = (allq[1] - allq[0]) * 1e10
sigma[i] = wavelength[i] ^ 2 * thickness / 2 / pi * np.sum(
allIq * allq * alldq)
s[i] = 1 - exp(-sigma)
dq = (q[1] - q[0]) * 1e10
a = (x < threshold)
acceptq = a * q
acceptIq = a * Iq
G[i] = np.sum(
besselj(0, acceptq * SElength_i) * acceptIq * acceptq * dq)
# G[i]=np.sum(integral)
G *= dq * 1e10 * 2 * pi
P = exp(thickness * wavelength**2 / (4 * pi**2) * (G - G[0]))
def analyticSolution(r, t, eps):
""" Method that calculates the analytic solution to the differential equation:
dw/dt = w'' + w/r, w = w(r,t), 0 < r < 1
where t is the time.
The accuracy can be controlled by changing the value of sumtol1, which depicts the
relative accuracy.
Args:
r(float): radial coordinat
t(float): time
eps(float): numerical tollerance
Returns:
w(float): velocity, us - ur
"""
if r == 1:
w = 0
elif r < eps and t < eps:
w = 1
else:
# calculate first element seperately
n = 1
lam1 = j0zero(n)
arg1 = r * lam1
term1 = np.exp(-t * lam1**2) * besselj(
0, arg1) / (besselj(1, lam1) * lam1**3)
sum1 = term1
sumtol = 1 * 10**-8
test = 1
while test > sumtol:
n = n + 1
lamn = j0zero(n)
arg = r * lamn
term = np.exp(-t * lamn**2) * besselj(
0, arg) / (besselj(1, lamn) * lamn**3)
sum1 = sum1 + term
test = np.abs(term / term1)
w = 8 * sum1
return w
def compute(x, rho, omega, mu):
R = 0.01
dpdx = -1
alpha = R * sqrt(rho * omega / mu)
numerator = besselj(0, 1j**1.5 * alpha * x / R)
denominator = besselj(0, 1j**1.5 * alpha)
bracket = 1 - numerator / denominator
coefficient = dpdx * 1j / (rho * omega)
profile = coefficient * bracket
profile = profile.real
maxValue = np.amax(profile)
profile = profile / maxValue
return profile
def analyticSolution(r,t,eps):
""" Method that calculates the analytic solution to the differential equation:
dw/dt = w'' + w/r, w = w(r,t), 0 < r < 1
where t is the time.
The accuracy can be controlled by changing the value of sumtol1, which depicts the
relative accuracy.
Args:
r(float): radial coordinat
t(float): time
eps(float): numerical tollerance
Returns:
w(float): velocity, us - ur
"""
if r == 1:
w = 0
elif r < eps and t < eps:
w = 1
else:
# calculate first element seperately
n = 1
lam1 = j0zero(n)
arg1 = r*lam1
term1 = np.exp(-t*lam1**2)*besselj(0,arg1)/(besselj(1,lam1)*lam1**3)
sum1 = term1
sumtol = 1*10**-8
test = 1
while test > sumtol:
n = n + 1
lamn = j0zero(n)
arg = r*lamn
term = np.exp(-t*lamn**2)*besselj(0,arg)/(besselj(1,lamn)*lamn**3)
sum1 = sum1 + term
test = np.abs(term/term1)
w = 8 * sum1
return w
def __calculate(args):
index_i, index_f, c, Bessel_Jn, Bessel_Jm, F, Nzeros, N = args
H = numpy.full((index_f - index_i), 0j)
for jj in range(index_i, index_f):
C = besselj(0, c[:Nzeros] * c[jj] / c[N]) / (Bessel_Jn * Bessel_Jm[jj])
H[jj - index_i] = numpy.dot(C[:Nzeros], F[:Nzeros])
return H
def Hankel_Transform_MGS(h, R, c, Nzeros=0, multipool=True):
N = len(h)
if Nzeros <= 0: Nzeros = N
V = c[N] / (2 * numpy.pi * R)
h = h.conjugate().T
m1 = (numpy.abs(besselj(1, c[0:N])) / R).conjugate().T
m2 = m1 * R / V
F = numpy.divide(h, m1)
Bessel_Jn = numpy.abs(besselj(1, c[0:Nzeros])) / (2 / c[N])
Bessel_Jm = numpy.abs(besselj(1, c))
if multipool:
args = __create_args(__N_CPU, c, Bessel_Jn, Bessel_Jm, F, Nzeros, N)
p = multiprocessing.Pool(__N_CPU)
H = numpy.concatenate(p.map(__calculate, args))
p.close()
else:
H = numpy.full(N, 0j)
for jj in range(0, N):
C = besselj(
0, c[0:Nzeros] * c[jj] / c[N]) / (Bessel_Jn * Bessel_Jm[jj])
H[jj] = numpy.dot(C[0:Nzeros], F[0:Nzeros])
H = H.conjugate().T
H = numpy.multiply(H, m2)
H = H.conjugate().T
return H
def hankel(SElength, wavelength, thickness, q, Iq):
r"""
Compute the expected SESANS polarization for a given SANS pattern.
Uses the hankel transform followed by the exponential. The values for *zz*
(or spin echo length, or delta), wavelength and sample thickness should
come from the dataset. $q$ should be chosen such that the oscillations
in $I(q)$ are well sampled (e.g., $5 \cdot 2 \pi/d_{\max}$).
*SElength* [A] is the set of $z$ points at which to compute the
Hankel transform
*wavelength* [m] is the wavelength of each individual point *zz*
*thickness* [cm] is the sample thickness.
*q* [A$^{-1}$] is the set of $q$ points at which the model has been
computed. These should be equally spaced.
*I* [cm$^{-1}$] is the value of the SANS model at *q*
"""
from sas.sascalc.data_util.nxsunit import Converter
wavelength = Converter(wavelength[1])(wavelength[0], "A")
thickness = Converter(thickness[1])(thickness[0], "A")
Iq = Converter("1/cm")(Iq,
"1/A") # All models default to inverse centimeters
SElength = Converter(SElength[1])(SElength[0], "A")
G = np.zeros_like(SElength, 'd')
#==============================================================================
# Hankel Transform method if "wavelength" is a scalar; mono-chromatic SESANS
#==============================================================================
for i, SElength_i in enumerate(SElength):
integral = besselj(0, q * SElength_i) * Iq * q
G[i] = np.sum(integral)
G0 = np.sum(Iq * q)
# [m^-1] step size in q, needed for integration
dq = (q[1] - q[0])
# integration step, convert q into [m**-1] and 2 pi circle integration
G *= dq * 2 * pi
G0 = np.sum(Iq * q) * dq * 2 * np.pi
P = exp(thickness * wavelength**2 / (4 * pi**2) * (G - G0))
return P
def hankel(SElength, wavelength, thickness, q, Iq):
r"""
Compute the expected SESANS polarization for a given SANS pattern.
Uses the hankel transform followed by the exponential. The values for *zz*
(or spin echo length, or delta), wavelength and sample thickness should
come from the dataset. $q$ should be chosen such that the oscillations
in $I(q)$ are well sampled (e.g., $5 \cdot 2 \pi/d_{\max}$).
*SElength* [A] is the set of $z$ points at which to compute the
Hankel transform
*wavelength* [m] is the wavelength of each individual point *zz*
*thickness* [cm] is the sample thickness.
*q* [A$^{-1}$] is the set of $q$ points at which the model has been
computed. These should be equally spaced.
*I* [cm$^{-1}$] is the value of the SANS model at *q*
"""
from sas.sascalc.data_util.nxsunit import Converter
wavelength = Converter(wavelength[1])(wavelength[0],"A")
thickness = Converter(thickness[1])(thickness[0],"A")
Iq = Converter("1/cm")(Iq,"1/A") # All models default to inverse centimeters
SElength = Converter(SElength[1])(SElength[0],"A")
G = np.zeros_like(SElength, 'd')
#==============================================================================
# Hankel Transform method if "wavelength" is a scalar; mono-chromatic SESANS
#==============================================================================
for i, SElength_i in enumerate(SElength):
integral = besselj(0, q*SElength_i)*Iq*q
G[i] = np.sum(integral)
G0 = np.sum(Iq*q)
# [m^-1] step size in q, needed for integration
dq = (q[1]-q[0])
# integration step, convert q into [m**-1] and 2 pi circle integration
G *= dq*2*pi
G0 = np.sum(Iq*q)*dq*2*np.pi
P = exp(thickness*wavelength**2/(4*pi**2)*(G-G0))
return P
def calc_ftk_svd(n_bessel, eps, pf_grid, tr_grid):
all_UU = [None] * (2 * n_bessel + 1)
all_SSVV = [None] * (2 * n_bessel + 1)
all_rnks = np.zeros(2 * n_bessel + 1, dtype=np.int32)
xnodesr = pf_grid['xnodesr']
all_delta = tr_grid['all_delta']
n_delta = tr_grid['n_delta']
n_omega = tr_grid['n_omega']
n_trans = tr_grid['n_trans']
for qp in range(-n_bessel, n_bessel + 1):
J_n = besselj(qp, -all_delta[:, np.newaxis] * xnodesr[np.newaxis, :])
U, S, Vh = np.linalg.svd(J_n)
ind = S > eps
rnk = sum(ind)
all_rnks[qp + n_bessel] = rnk
all_UU[qp + n_bessel] = U[:, :rnk]
all_SSVV[qp + n_bessel] = S[:rnk, np.newaxis] * Vh[:rnk, :]
SSVV_big = np.concatenate(all_SSVV, axis=0)
UUU = np.concatenate(all_UU, axis=1)
all_omega = np.concatenate(
[2 * np.pi / n_om * np.arange(n_om) for n_om in n_omega if n_om > 0])
all_qp = np.concatenate([(k - n_bessel) * np.ones(n)
for k, n in enumerate(all_rnks)])
vec_omega = np.exp(1j * all_qp[np.newaxis, :] *
(all_omega[:, np.newaxis] - np.pi / 2))
BigMul_left = np.zeros((sum(all_rnks), n_trans), dtype=np.complex128)
cnt = 0
for kk in range(n_delta):
n_om = n_omega[kk]
BigMul_left[:, cnt:cnt + n_om] = (UUU[kk, :][np.newaxis, :].T *
vec_omega[cnt:cnt + n_om, :].T)
cnt += n_om
return all_rnks, BigMul_left, SSVV_big
def pointils2(band,wave):
#VERSION 2 updates for replanned optics (smaller grating footprint)
gratingsizes = np.array([81., 81., 84.4, 84.4])
#make function to generate pointils.
#convolution of grating function with airy function for the relevant band
deltawave = 1e-6
[sigma,alpha,beta0,order,fcam] = get_geocarb_gratinginfo(band)
gratingsize=gratingsizes[band]
#find central wavelength
cenwave = 0.5*(wave[len(wave)//2]+wave[len(wave)//2+1])#gratinglambda(sigma,alpha,beta0,m=order)
#wave=np.arange(0.001*2/deltawave)*deltawave+cenwave-0.001
#compute beta angles for these wavelengths
betas = betaangle(wave,sigma,alpha,m=order)
#FIRST DO GRATING FUNCTION
#number of illuminated grooves
Ngrooves = gratingsize*1000./sigma
#phase shift
deltaphi = 2*np.pi*sigma/cenwave*(np.sin(betas*dtor)-np.sin(beta0*dtor))
#total phase shift across grating
phi = Ngrooves*deltaphi
inten = 1/Ngrooves**2*(np.sin(phi/2)/np.sin(deltaphi/2))**2
deltawave = wave-cenwave
#NOW FOR AIRY FUNCTION
k = 2*np.pi/cenwave
ap = 75./2./2. #radius of aperture in mm (extra factor of two from descope)
bx = k*ap*1000.*np.sin((betas-beta0)*dtor)
#take into account that beam speed in spectral direction
#has changed due to grating magnification
bx = bx*np.cos(beta0*dtor)/np.cos(alpha*dtor)
airy = (2*besselj(1,bx)/bx)**2
#pdb.set_trace()
airy = airy/np.nanmax(airy)
#diffraction limit FWHM
diffFWHM = cenwave*3.2*np.sqrt(2)*np.cos(alpha*dtor)/np.cos(beta0*dtor)
#POINT ILS IS CONVOLUTION OF GRATING FUNCTION WITH AIRY FUNCTION
pointils = convolve1d(inten,airy, mode='constant', cval=0.0)
#pdb.set_trace()
pointils = pointils/pointils.max()
return pointils
def _LvN_propagate(self, S, NAC, dt):
nexp = 7
Heff = S - ii * NAC
self._LvN_evals(Heff)
R = dt * (self.e_max - self.e_min) / 2
G = dt * self.e_min
X = -ii * self.dt * Heff / R
c = np.full((nexp, ), 2)
c[0] = 1
phi = [self.rho, np.dot(X, self.rho) - np.dot(self.rho, X)]
for k in range(2, nexp):
phi.append(2 * (np.dot(X, phi[k - 1]) - np.dot(phi[k - 1], X)) +
phi[k - 2])
a = [np.exp(ii * (R + G)) * c[k] * besselj(k, R) for k in range(nexp)]
rho_dt = np.zeros_like(self.rho)
for k in range(nexp):
rho_dt += a[k] * phi[k]
self.rho = rho_dt
def calculateEigenPair(a, c0, f, azimuthal_m, radial_n, mode_type, sign=1):
if mode_type not in {"Kelvin", "Poincare"}:
raise ValueError("Unknown mode_type: '" + mode_type + "'")
if sign not in {-1, 1}:
raise ValueError("Invalid sign: '" + str(sign) + "'")
if mode_type == 'Kelvin' and sign == 1:
raise ValueError(
"Invalid parameter selection. Kelvin modes with clockwise phase propagation do not exist"
)
bracket_factor = 2e-2
eps = 1e-7
PoincareEqn = lambda sig: (a / np.sqrt(c0**2 / (sig**2 - f**2))) * (
(1 / azimuthal_m) * besselj(azimuthal_m - 1, a / np.sqrt(c0**2 / (
sig**2 - f**2))) /
(a / np.sqrt(c0**2 / (sig**2 - f**2))) * besselj(
azimuthal_m, a / np.sqrt(c0**2 / (sig**2 - f**2)))) - 1 + (f / sig)
KelvinEqn = lambda sig: (a / np.sqrt(c0**2 / (f**2 - sig**2))) * (
(1 / azimuthal_m) * besseli(azimuthal_m - 1, a / np.sqrt(c0**2 / (
f**2 - sig**2))) /
(a / np.sqrt(c0**2 / (f**2 - sig**2))) * besseli(
azimuthal_m, a / np.sqrt(c0**2 / (f**2 - sig**2)))) - 1 + (f / sig)
# Calculate sigma
if mode_type == 'Poincare' and sign == 1:
sig = f
for n in range(0, radial_n):
rt = root(PoincareEqn,
method="bisect",
bracket=[(1 + eps) * sig, (1 + bracket_factor) * sig])
if rt.root is None:
print(
"Did not find an eigenvalue in expected range for Poincare mode with n="
+ str(n))
sig = rt.root
elif mode_type == 'Poincare' and sign == -1:
sig = -f
for n in range(0, radial_n):
rt = root(PoincareEqn,
method="bisect",
bracket=[(1 + bracket_factor) * sig, (1 + eps) * sig])
if rt.root is None:
print(
"Did not find an eigenvalue in expected range for Poincare mode with n="
+ str(n))
sig = rt.root
else:
sig = -f
for n in range(
0, radial_n
): # Can also restrict n's on kelvin mode using Csanady's restriction on S.
rt = root(KelvinEqn,
method="bisect",
bracket=[(1 - eps) * sig, -eps])
if rt.root is None:
print(
"Did not find an eigenvalue in expected range for Kelvin mode with n="
+ str(n))
sig = rt.root
Theta = lambda theta: np.exp(1j * azimuthal_m * theta)
if mode_type == "Kelvin":
R = lambda r: besseli(azimuthal_m,
np.sqrt((f**2 - sig**2) / c0**2) * r)
else:
#Poincare
R = lambda r: besselj(azimuthal_m,
np.sqrt((sig**2 - f**2) / c0**2) * r)
eigenfunction = lambda r, theta: R(r) * Theta(theta)
return (sig, (lambda r, theta: eigenfunction(r, theta)))
def find_boundary_data(f, dt, xbdy, ybdy, ν, J, κ, M, k_i, k_e):
'''
Computes u and du/dv on the boundary by solving the BIE system
Attributes
======
kernel of the double layer, L
kernel of the single layer, M
solution on the boundary, u
normal derivative on the boundary, dvu
'''
E_C = 0.5772156649015328606065120900824; # Euler's constant
N = M / 2;
m = np.arange(1, N);
k_ratio = k_e/k_i
# Create array for ifft...
a = [0]
a.extend(1/m)
a.append(1/N)
a.extend((1/m)[::-1])
Rj = -2 * np.pi * np.fft.ifft(a);
R = np.real(toeplitz(Rj, Rj));
# Prepare arrays for solving integral equation of the boundary (Kress pt. 2)
L_e = np.zeros((M, M), dtype=complex);
M_e = np.zeros((M, M), dtype=complex);
L_i = np.zeros((M, M), dtype=complex);
M_i = np.zeros((M, M), dtype=complex);
L1_e = np.zeros((M, M), dtype=complex);
L2_e = np.zeros((M, M), dtype=complex);
M1_e = np.zeros((M, M), dtype=complex);
M2_e = np.zeros((M, M), dtype=complex);
L1_i = np.zeros((M, M), dtype=complex);
L2_i = np.zeros((M, M), dtype=complex);
M1_i = np.zeros((M, M), dtype=complex);
M2_i = np.zeros((M, M), dtype=complex);
A = np.zeros((2*M, 2*M), dtype=complex);
F = np.zeros(2*M, dtype=complex);
for m in range(0, M):
for n in range(0, M):
rdiff = np.asarray([xbdy[m] - xbdy[n], ybdy[m] - ybdy[n]]);
distance = sqrt(np.power(rdiff[:][0], 2) + np.power(rdiff[:][1], 2));
M1_e[m][n] = -0.5 / np.pi * J[n] * besselj(0, k_e * distance);
M1_i[m][n] = -0.5 / np.pi * J[n] * besselj(0, k_i * distance);
if m == n:
#Implementration of the Kress Quadrature
L1_e[m][n] = 0;
L2_e[m][n] = 0.5 / np.pi * κ[n] * J[n];
M2_e[m][n] = J[n] * ( 0.5 * 1j - E_C / np.pi - 0.5 / np.pi * np.log( 0.25 * np.power(k_e, 2) * np.power(J[n], 2)));
L1_i[m][n] = 0;
L2_i[m][n] = 0.5 / np.pi * κ[n] * J[n];
M2_i[m][n] = J[n] * ( 0.5 * 1j - E_C / np.pi - 0.5 / np.pi * np.log( 0.25 * np.power(k_i, 2) * np.power(J[n], 2)));
else:
#Compute cosθ and logterm for kernel calculation
a = np.asmatrix([ν[0][n], ν[1][n]]);
b = np.asmatrix(rdiff/distance);
cosθ = np.asscalar(np.matmul(a, b.transpose()));
logterm = log(float(4 * np.power(sin(0.5 * (m - n) * dt), 2)));
#Compute kernels for the BIE system
L1_e[m][n] = 0.5 * k_e / np.pi * J[n] * cosθ * besselj(1, k_e * distance);
L2_e[m][n] = 0.5 * 1j * k_e * J[n] * cosθ * hankel1(1, k_e * distance) - L1_e[m][n] * logterm;
M2_e[m][n] = 0.5 * 1j * J[n] * hankel1(0, k_e * distance) - M1_e[m][n] * logterm;
L1_i[m][n] = 0.5 * k_i / np.pi * J[n] * cosθ * besselj(1, k_i * distance);
L2_i[m][n] = 0.5 * 1j * k_i * J[n] * cosθ * hankel1(1, k_i * distance) - L1_i[m][n] * logterm;
M2_i[m][n] = 0.5 * 1j * J[n] * hankel1(0, k_i * distance) - M1_i[m][n] * logterm;
L_e[m][n] = 0.5 * R[m][n] * (L1_e[m][n]) + 0.5 * (np.pi / N) * (L2_e[m][n]);
M_e[m][n] = (0.5 * R[m][n] * (M1_e[m][n]) + 0.5 * (np.pi / N) * (M2_e[m][n]));
L_i[m][n] = 0.5 * R[m][n] * (L1_i[m][n]) + 0.5 * (np.pi / N) * (L2_i[m][n]);
M_i[m][n] = 0.5 * R[m][n] * (M1_i[m][n]) + 0.5 * (np.pi / N) * (M2_i[m][n]);
def find_boundary_data(f, B, M, k_i, k_e):
'''
Computes the boundary data (trace u and normal derivative dvu)
by solving the BIE system using Kress quadrature:
(I/2 - De) u + (k_e/k_i Se) dvu = f
(I/2 + Di) u - Si dvu = 0
Parameters
==========
f: source (array)
B: boundary, lambdified (class)
M: number of quadrature points on the boundary (even integer)
k_i: wavenumber interior domain (complex number)
k_e: wavenumber exterior domain (complex number)
Returns
==========
u: solution on the boundary (array)
dvu: normal derivative of the solution on the boundary (array)
'''
#Define the boundary features
dt = 2.0 * np.pi / (M)
θ = np.arange(0, 2 * np.pi, dt)
x = B.y_l(θ)[0][:]
y = B.y_l(θ)[1][:]
ν = B.ν_l(θ)
J = B.J_l(θ)
κ = B.κ_l(θ)
#Define other parameters
k_ratio = k_e / k_i
E_C = np.euler_gamma # Euler's constant
N = M / 2
m = np.arange(1, N)
# Create array for ifft
a = [0]
a.extend(1 / m)
a.append(1 / N)
a.extend((1 / m)[::-1])
Rj = -2 * np.pi * np.fft.ifft(a)
R = np.real(toeplitz(Rj, Rj))
# Prepare arrays for solving integral equation of the boundary (Kress pt. 2)
A = np.zeros((2 * M, 2 * M), dtype=complex)
F = np.zeros(2 * M, dtype=complex)
cosθ_mn = np.zeros((M, M), dtype=complex)
logterm_mn = np.zeros((M, M), dtype=complex)
besselj_0e_mn = np.zeros((M, M), dtype=complex)
besselj_1e_mn = np.zeros((M, M), dtype=complex)
hankel1_0e_mn = np.zeros((M, M), dtype=complex)
hankel1_1e_mn = np.zeros((M, M), dtype=complex)
besselj_0i_mn = np.zeros((M, M), dtype=complex)
besselj_1i_mn = np.zeros((M, M), dtype=complex)
hankel1_0i_mn = np.zeros((M, M), dtype=complex)
hankel1_1i_mn = np.zeros((M, M), dtype=complex)
logterm_o = {}
for i in range(-M, M):
if i != 0:
logterm_o[i] = log(float(4 * np.power(np.sin(0.5 * i * dt), 2)))
for m in range(0, M):
for n in range(0, M):
rdiff = np.asarray([x[m] - x[n], y[m] - y[n]])
r_distance = sqrt(rdiff[:][0]**2 + rdiff[:][1]**2)
besselj_0e_mn[m, n] = besselj(0, k_e * r_distance)
besselj_0i_mn[m, n] = besselj(0, k_i * r_distance)
if m != n:
cosθ_mn[m, n] = ν[0][n] * (rdiff[0] / r_distance) + ν[1][n] * (
rdiff[1] / r_distance)
logterm_mn[m, n] = logterm_o[m - n]
besselj_1e_mn[m, n] = besselj(1, k_e * r_distance)
hankel1_1e_mn[m, n] = hankel1(1, k_e * r_distance)
hankel1_0e_mn[m, n] = hankel1(0, k_e * r_distance)
besselj_1i_mn[m, n] = besselj(1, k_i * r_distance)
hankel1_1i_mn[m, n] = hankel1(1, k_i * r_distance)
hankel1_0i_mn[m, n] = hankel1(0, k_i * r_distance)
#Compute kernels of exterior of boundary
L1_e = 0.5 * k_e / np.pi * J * cosθ_mn * besselj_1e_mn
L2_e = 0.5 * 1j * k_e * J * cosθ_mn * hankel1_1e_mn - L1_e * logterm_mn
M1_e = -0.5 / np.pi * J * besselj_0e_mn
M2_e = 0.5 * 1j * J * hankel1_0e_mn - M1_e * logterm_mn
# Usage of Kress Quadrature
diag = np.arange(0, M)
M2_e[diag,
diag] = J * (0.5 * 1j - E_C / np.pi - 0.5 / np.pi * np.log(0.25 *
(k_e**2) *
(J**2)))
L1_e[diag, diag] = 0
L2_e[diag, diag] = 0.5 / np.pi * κ * J
L_e = 0.5 * R * L1_e + 0.5 * (np.pi / N) * L2_e
M_e = 0.5 * R * M1_e + 0.5 * (np.pi / N) * M2_e
#Calculate kernels of interior of boundary
L1_i = 0.5 * k_i / np.pi * J * cosθ_mn * besselj_1i_mn
L2_i = 0.5 * 1j * k_i * J * cosθ_mn * hankel1_1i_mn - L1_i * logterm_mn
M1_i = -0.5 / np.pi * J * besselj_0i_mn
M2_i = 0.5 * 1j * J * hankel1_0i_mn - M1_i * logterm_mn
#Replace diagonals where m=n with exception cases
L1_i[diag, diag] = 0
L2_i[diag, diag] = 0.5 / np.pi * κ * J
M2_i[diag,
diag] = J * (0.5j - E_C / np.pi - 0.5 / np.pi * np.log(0.25 *
(k_i**2) *
(J**2)))
L_i = 0.5 * R * L1_i + 0.5 * (np.pi / N) * L2_i
M_i = 0.5 * R * M1_i + 0.5 * (np.pi / N) * M2_i
#Matrix of combined representation formulas
A11 = 0.5 * np.identity(M) - L_e
A12 = k_ratio * M_e
A21 = 0.5 * np.identity(M) + L_i
A22 = -M_i
#Combine A's into one matrix, A
A[0:M, 0:M] = A11
A[0:M, M:2 * M] = A12
A[M:2 * M, 0:M] = A21
A[M:2 * M, M:2 * M] = A22
#Make matrix for f and 0's, call it F
F[0:M] = f
#Solve for u and dvu from AU = F
U = np.linalg.solve(A, F)
u = U[0:M]
dvu = U[M:2 * M]
return u, dvu
def fanNoise(self, S, theta, Thrust, power_Watts):
"""
Inputs
1.S - distance from observer location to noise source [m]
2.theta - angle from thrust direction to observer location [rad]
2.Thrust - thrust [N]
3.power_watts - power in watts, from a SINGLE rotor
Calculates and stores:
spl - Sound pressure level in dB
"""
NtoLb = 0.224809 # Newtons to lb
WtoHp = 0.00134102 # Watts to hp
# m2f = 1.0/0.3048 # meters to feet
Pref = 0.0002 # dynes/cm^2
k = 6.1e-27 # proportionality constant
# Average blade CL
CL = 6 * self.ctsigma # Ct / geom.solidity
# Thrust in pounds
T_lb = Thrust * NtoLb
# Parameters
VTip = self.tipspeed # tip speed, m/s
S_ft = S * m2f # Distance to source in ft
R_ft = self.radius * m2f # Radius in ft
A_ft2 = self.area * m2f * m2f # Disc area in ft^2
P_Hp = power_Watts * WtoHp # Power in Hp
M_t = VTip / self.ainf # tip mach number
A_b = A_ft2 * self.solidity # blade area, sq. ft
# print(self.power*0.001)
# print(self.solidity)
# print(S_ft, R_ft, A_ft2, P_Hp, M_t, A_b)
# quit()
# Compute rotational noise
m_max = 10 # Maximum harmonic number
p_m = numpy.zeros(m_max)
for m in range(1, m_max + 1):
p_m[m-1] = 169.3 * m * self.nblade * R_ft * M_t / (S_ft * A_ft2) * \
(0.76 * P_Hp / M_t**2 - T_lb * cos(theta)) * \
besselj(m * self.nblade, 0.8 * M_t * m * self.nblade * sin(theta))
# Compute total RMS sound pressure level in dynes/cm^2 = 0.1 Pa
p = numpy.sqrt(numpy.sum(numpy.square(p_m)))
# Convert to SPL in dB's
SPL_rotational = 20 * numpy.log10(p / Pref)
# Vortex Noise (pg. 11)
# SPL = 10 * log(k * A_b * V_0.7^6 / 10^-16) + 20 * log(CL/0.4) [dB at 300 ft]
# where
# k = constant of proportionality = 6.1e-27
# k/1e-16 = 6.1e-11
# A_b = propeller blade area [ft^2]
# V_0.7 = velocity at 0.7 * radius
# SPL_vortex = 10 * log10(k * A_b * (0.7 * VTip)**6 / 1e-16) + 20 * log10(CL / 0.4)
SPL_vortex = 10 * log10(6.1e-11 * A_b *
(0.7 * VTip)**6) + 20 * log10(CL / 0.4)
# Total noise
# Adding atmospheric attenuation of 6 dB for every doubling of distance from reference
spl = 10 * log10(10**(SPL_rotational / 10) + 10**
(SPL_vortex / 10)) - 6 * log2(S_ft / 300)
return spl
def getlhs(W,M): return M
# define the PDE
pde = pde.Parabolic(NComponents = 2, Mesh = mesh, StiffMat = a.globalStiffMat, MassMat = a.globalMassMat)
pde.setNeumannFunc = gNeumann
pde.setSrcFunc = fsrc
pde.AssembBlockStiffMat = makeBlockMat
pde.AssembBlockMassMat = makeBlockMat
pde.AssembLHS = getlhs
import linsolv
W_free = makeBlockMat(pde.getFreeNodeArray(W))
# initial condition (bessel perturbation)
#J_m (k_mn * r) * Cos(theta) written in cartesian co-ordinates
f0 = [lambda p: c_10 + 0.03*c_10 * besselj(m, k_mn * np.sqrt(p[0]**2 + p[1]**2)) * p[0]/(1e-4 + np.sqrt(p[0]**2 + p[1]**2)),
lambda p: c_20 + 0.03*c_20 * besselj(m, k_mn * np.sqrt(p[0]**2 + p[1]**2)) * p[0]/(1e-4 + np.sqrt(p[0]**2 + p[1]**2))]
u0 = np.zeros([mesh.NumNodes,2])
for i, node in enumerate(mesh.Nodes):
u0[i,0] = f0[0](node)
u0[i,1] = f0[1](node)
# start time loop
np.savetxt(outfile_fmt % 0, u0)
dt = 1e-4
print "dt = ", dt
print "Mesh size = ", mesh.Diam
raw_input('Press any key to start time loop')
def __inner_loop_single_iteration(bessel_zeros_jj, bessel_jm_jj, bessel_zeros_nz, bessel_zeros_n, bessel_jn, f_nz):
return numpy.dot(besselj(0, bessel_zeros_nz * bessel_zeros_jj / bessel_zeros_n) / (bessel_jn * bessel_jm_jj),
f_nz)
clf()
subplot(211) # plot the SANS calculation
plot(q, I, 'k')
loglog(q, I)
xlim([0.01, 1])
ylim([1, 1e9])
xlabel(r'$Q [nm^{-1}]$')
ylabel(r'$d\Sigma/d\Omega [m^{-1}]$')
# Hankel transform to nice range for plot
nz = 61
zz = linspace(0, 240, nz)
# [nm], should be less than reciprocal from q
G = zeros(nz)
for i in range(len(zz)):
integr = besselj(0, q * zz[i]) * I * q
G[i] = sum(integr)
G = G * dq * 1e9 * 2 * pi
# integr step, conver q into [m**-1] and 2 pi circle integr
# plot(zz,G);
stt = th * Lambda**2 / 4 / pi / pi * G[
0] # scattering power according to SANS formalism
PP = exp(th * Lambda**2 / 4 / pi / pi * (G - G[0]))
subplot(212)
plot(zz, PP, 'k', label="Hankel transform") # Hankel transform 1D
xlabel('spin-echo length [nm]')
ylabel('polarisation normalised')
hold(True)
# Cosine transformation of 2D scattering patern
opts = dict()
# --------------------------------------------------------------------------
# find phase factors
opts["criteria"] = criteria
max_order = ceil(1.4 * tau + np.log(1e14))
if np.mod(max_order, 2) == 1:
max_order -= 1
# --------------------------------------------------------------------------
# even part
coeff = np.zeros((max_order // 2 + 1, 1))
for i in range(len(coeff)):
coeff[i] = (-1)**(i) * besselj(2 * i, tau)
coeff[0] /= 2
[phi1, out1] = QSP_Solver(coeff, 0, opts)
print("- Info: \t\tQSP phase factors --- solved by L-BFGS\n")
print("- Parity: \t\t%s\n- Degree: \t\t%d\n", "even", max_order)
print("- Iteration times: \t%d\n", out1["iter"])
print("- CPU time: \t%.1f s\n", out1["time"])
#--------------------------------------------------------------------------
# odd part
coeff = np.zeros((max_order / 2 + 1, 1))
for i in range(len(coeff)):
coeff[i] = (-1)**(i) * besselj(2 * i + 1, tau)
clf()
subplot(211) # plot the SANS calculation
plot(q,I,'k')
loglog(q,I)
xlim([0.01, 1])
ylim([1, 1e9])
xlabel(r'$Q [nm^{-1}]$')
ylabel(r'$d\Sigma/d\Omega [m^{-1}]$')
# Hankel transform to nice range for plot
nz=61;
zz=linspace(0,240,nz); # [nm], should be less than reciprocal from q
G=zeros(nz);
for i in range(len(zz)):
integr=besselj(0,q*zz[i])*I*q;
G[i]=sum(integr);
G=G*dq*1e9*2*pi; # integr step, conver q into [m**-1] and 2 pi circle integr
# plot(zz,G);
stt= th*Lambda**2/4/pi/pi*G[0] # scattering power according to SANS formalism
PP=exp(th*Lambda**2/4/pi/pi*(G-G[0]));
subplot(212)
plot(zz,PP,'k',label="Hankel transform") # Hankel transform 1D
xlabel('spin-echo length [nm]')
ylabel('polarisation normalised')
hold(True)
# Cosine transformation of 2D scattering patern
if False:
qy,qz = meshgrid(q,q)
def compute_oof(self, array, radii):
array = array.astype(np.double)
shape = array.shape
output = np.zeros(shape)
self.check_normalization(radii)
imgfft = fft(array)
x, y, z, sphere_radius = get_min_sphere_radius(shape, self.spacing)
for radius in tqdm(radii):
tqdm.write(f'Computing radius {radius:.3f}...')
circle = circle_length(radius)
nu = 1.5
z = circle * EPSILON
bessel = besselj(nu, z) / EPSILON**(3 / 2)
base = radius / np.sqrt(2 * radius * self.sigma - self.sigma**2)
exponent = self.normalization_type
volume = get_sphere_volume(radius)
normalization = volume / bessel / radius**2 * base**exponent
exponent = -self.sigma**2 * 2 * np.pi**2 * sphere_radius**2
num = normalization * np.exp(exponent)
den = sphere_radius**(3 / 2)
besselj_buffer = num / den
cs = circle * sphere_radius
a = np.sin(cs) / cs - np.cos(cs)
b = np.sqrt(1 / (np.pi**2 * radius * sphere_radius))
besselj_buffer = besselj_buffer * a * b * imgfft
outputfeature_11 = np.real(ifft(x * x * besselj_buffer))
outputfeature_12 = np.real(ifft(x * y * besselj_buffer))
outputfeature_13 = np.real(ifft(x * z * besselj_buffer))
outputfeature_22 = np.real(ifft(y * y * besselj_buffer))
outputfeature_23 = np.real(ifft(y * z * besselj_buffer))
outputfeature_33 = np.real(ifft(z * z * besselj_buffer))
eigenvalues = eigenvalue_field33(
outputfeature_11, outputfeature_12, outputfeature_13,
outputfeature_22, outputfeature_23, outputfeature_33)
lambda_1, lambda_2, lambda_3 = eigenvalues
maxe = np.copy(lambda_1)
mine = np.copy(lambda_1)
mide = maxe + lambda_2 + lambda_3
if self.use_absolute:
maxe[np.abs(lambda_2) > np.abs(maxe)] = lambda_2[
np.abs(lambda_2) > np.abs(maxe)]
mine[np.abs(lambda_2) < np.abs(mine)] = lambda_2[
np.abs(lambda_2) < np.abs(mine)]
maxe[np.abs(lambda_3) > np.abs(maxe)] = lambda_3[
np.abs(lambda_3) > np.abs(maxe)]
mine[np.abs(lambda_3) < np.abs(mine)] = lambda_3[
np.abs(lambda_3) < np.abs(mine)]
else:
maxe[lambda_2 > np.abs(maxe)] = lambda_2[
lambda_2 > np.abs(maxe)]
mine[lambda_2 < np.abs(mine)] = lambda_2[
lambda_2 < np.abs(mine)]
maxe[lambda_3 > np.abs(maxe)] = lambda_3[
lambda_3 > np.abs(maxe)]
mine[lambda_3 < np.abs(mine)] = lambda_3[
lambda_3 < np.abs(mine)]
mide -= maxe + mine
if self.response_type == 0:
tmpfeature = maxe
elif self.response_type == 1:
tmpfeature = maxe + mide
elif self.response_type == 2:
tmpfeature = np.sqrt(np.maximum(0, maxe * mide))
elif self.response_type == 3:
tmpfeature = np.sqrt(
np.maximum(0, maxe * mide) * np.maximum(0, mide))
elif self.response_type == 4:
tmpfeature = np.maximum(0, maxe)
elif self.response_type == 5:
tmpfeature = np.maximum(0, maxe + mide)
stronger_response = np.abs(tmpfeature) > np.abs(output)
output[stronger_response] = tmpfeature[stronger_response]
return output
else:
dia = float(input('Diameter of driver in mm : '))
f = float(input('Frequency of interested in Hz : '))
#
c = 345.0
rho_a = 1.2
phi = arange(-1 * pi / 2.0, pi / 2.0 + pi / 1800.0, pi / 1800.0)
phi = phi[
1:
1800] # Is this a bug? it's shifted to the right, past the end of the array.
#
a = dia / 1000.0
theta = arange(pi / 1800.0, pi, pi / 1800.0)
k = (2 * pi * f) / c
#
dir = 2 * divide(besselj(k * a, sin(theta)), (k * a * sin(theta)))
#
p = divide(dir, max(dir))
q = 20.0 * log10(p)
#
#p_s=[phi' p'];
q_s = (transpose(phi), transpose(q))
#
#Dirplot(q_s(:,1)*180/pi,q_s(:,2),'r-',[0 -30 5]);
ax = plt.subplot('111', projection='polar')
ax.plot(q_s[0], q_s[1], 'r-')
ax.set_rticks(range(-30, 1, 6))
ax.set_theta_offset(pi / 2.0)
ax.set_thetalim(thetamin=90, thetamax=-90)
ax.set_theta_direction(-1)
