def TacC2_calc(k=None,p=None,fDc=None,fMag=None,a=None,*args,**kwargs): C=fDc M=fMag s21=np.sinh(k * (1 - a)) s20=np.cosh(k * (1 - a)) s19=4 * M * k * p ** 2 * s21 s18=4 * M * k ** 2 * p ** 2 * s20 s17=4 * C * k ** 3 s16=2 * k ** 4 * p s15=k - 2 * a * k s14=8 * C * a * p ** 3 s13=2 * C * k * p ** 3 s12=4 * C * k * p ** 2 s11=4 * C * k ** 2 * p s10=8 * C * a * k ** 2 * p s9=4 * M * a * k * p ** 3 s8=2 * k ** 2 * p ** 3 s7=4 * C * p ** 3 s6=4 * M * k ** 2 * p s5=8 * C * k ** 2 s4=8 * C * p ** 2 s3=4 * M * k * p ** 2 s2=2 * C * k ** 2 * p ** 2 s1=4 * M * a * k ** 2 * p ** 2 line1=(s1 + s2) * np.cos(a * p) * np.cosh(a * k) + (s3 - s12 - s17) * np.cos(a * p) * np.sinh(k) + (- s12 - s3) * np.cos(a * p) * np.sinh(a * k) + (- 2 * M * k ** 2 * p ** 2 + s5 + s4) * np.cos(a * p) * np.cosh(k) line2=(2 * C * k ** 2 * p ** 2 * s20 - s4 - 4 * C * k * p ** 2 * s21 - s19 - s5 + s18 - 2 * M * k ** 2 * p ** 2 * np.cosh(s15) - 4 * M * a * k ** 2 * p ** 2 * s20) * np.cos(a * p) + (s6 - s7) * np.sin(a * p) * np.cosh(a * k) line3=(s13 + 2 * C * k ** 3 * p + 2 * M * k * p ** 3 + 4 * M * k ** 3 * p - 4 * C * a * k * p ** 3 - 4 * C * a * k ** 3 * p - s9 - 4 * M * a * k ** 3 * p) * np.sin(a * p) * np.sinh(k) + (s13 + s9) * np.sin(a * p) * np.sinh(a * k) line4=(s14 - s7 - s11 - s6 + s10) * np.sin(a * p) * np.cosh(k) + (s7 + 4 * C * p ** 3 * s20 - s14 + s11 + s6 - s10 - 4 * M * k ** 2 * p * s20 - 2 * C * k * p ** 3 * s21 - 4 * M * k * p ** 3 * s21 + 2 * M * k * p ** 3 * np.sinh(s15) + 4 * M * a * k * p ** 3 * s21) * np.sin(a * p) line5=(s17 + 8 * C * k * p ** 2 + s3) * np.sinh(k) - s3 * np.sinh(a * k) + (s1 - s4 - s2 - 4 * M * k ** 2 * p ** 2 - s5) * np.cosh(k) + s5 + s4 - s2 - s19 + s18 - s1 denom=(k ** 5 * p + k ** 3 * p ** 3) * np.sinh(k) + (- s16 - s8) * np.cosh(k) + s16 + s8 TacC2=(line1 + line2 + line3 + line4 + line5) / denom return TacC2
def _get_mean(self, sites, C, ln_y_ref, exp1, exp2, v1):
"""
Add site effects to an intensity.
Implements eq. 5
"""
# we do not support estimating of basin depth and instead
# rely on it being available (since we require it).
z1pt0 = sites.z1pt0
# we consider random variables being zero since we want
# to find the exact mean value.
eta = epsilon = 0
ln_y = (
# first line of eq. 13b
ln_y_ref + C['phi1'] *
np.log(np.clip(sites.vs30, -np.inf, v1) / 1130)
# second line
+ C['phi2'] * (exp1 - exp2)
* np.log((np.exp(ln_y_ref) + C['phi4']) / C['phi4'])
# third line
+ C['phi5']
* (1.0 - 1.0 / np.cosh(
C['phi6'] * (z1pt0 - C['phi7']).clip(0, np.inf)))
+ C['phi8'] / np.cosh(0.15 * (z1pt0 - 15).clip(0, np.inf))
# fourth line
+ eta + epsilon
)
return ln_y
def computeNextCom(self,p0,x0=[[0,0] , [0,0]],t=0.05):
px=p0[0]
py=p0[1]
'''Compute COM at time (t < durrationOfStep*(1-alpha) ) This function is
usefull for MPC implementation '''
#TODO check t < durrationOfStep*(1-alpha)
w2= self.g/self.h
w = np.sqrt(w2)
durrationOfStep = self.durrationOfStep
x0_x=np.matrix([[x0[0][0]],
[x0[0][1]]])
x0_y=np.matrix([[x0[1][0]],
[x0[1][1]]])
c0_x =x0_x[0,0]
c0_y =x0_y[0,0]
d_c0_x=x0_x[1,0]
d_c0_y=x0_y[1,0]
c_x = (c0_x -px) * np.cosh(w*t) + (d_c0_x/w) * np.sinh(w*t)+px
d_c_x = w*(c0_x -px) * np.sinh(w*t) + d_c0_x * np.cosh(w*t)
c_y = (c0_y -py) * np.cosh(w*t) + (d_c0_y/w) * np.sinh(w*t)+py
d_c_y = w*(c0_y -py) * np.sinh(w*t) + d_c0_y * np.cosh(w*t)
return [c_x , c_y , d_c_x , d_c_y]
def _fenton1985_dispersion_step(w,k,H,d):
""" Solve eqn 23 in Fenton1985, assuming no mean current (i.e., c_E = 0) using Newton's method.
NOTE: This approach was found to be numerically unstable unless a relaxation factor was applied at each iteration; however this required > 100 iterations for convergence in the cases tested.
"""
kd = k*d
S = 1./np.cosh(2*kd)
C0 = np.tanh(kd)**0.5
dC0 = d/(2*C0*np.cosh(kd)**2)
C2expr = (2.+7.*S**2)/(4.*(1.-S)**2)
C2 = C0 * C2expr
t2kd = np.tanh(2*kd)
dC2 = \
dC0 * C2expr \
+ C0 *( -7.*d*t2kd*S**2/(1.-S)**2 - d*t2kd*(7*S**2+2.)*S/(1.-S)**3 )
C4expr = (4. + 32.*S - 116.*S**2 - 400.*S**3 - 71.*S**4 + 146.*S**5) / (32*(1.-S)**5)
C4 = C0 * C4expr
dC4 = \
dC0 * C4expr \
+ C0 * ( -1460*d*t2kd*S**5 + 568*d*t2kd*S**4 \
+ 2400*d*t2kd*S**3 + 464*d*t2kd*S**2 - 64*d*t2kd*S ) / (32*(1.-S)**5) \
- C0 * ( 5*d*t2kd*S*(146*S**5-71*S**4-400*S**3-116*S**2+32*S+4) ) \
/ (16*(1.-S)**6)
kH_2 = k*H/2.
F = -w/(g*k)**0.5 + C0 + kH_2**2*C2 + kH_2**4*C4
J = dC0 + H*kH_2*C2 + kH_2**2*dC2 + 2*H*kH_2**3*C4 + kH_2**4*dC4
if DEBUG:
print ' C0:',C0
print ' C2:',C2
print ' C4:',C4
print 'dC0:',dC0
print 'dC2:',dC2
print 'dC4:',dC4
return F,J
def test_abcd_lossy_line(self):
'''
Lossy transmission line of characteristic impedance Z0, length l
and propagation constant gamma = alpha + j beta
○---------○
○---------○
has ABCD matrix of the form:
[ cosh(gamma l) Z0 sinh(gamma l) ]
[ 1/Z0 sinh(gamma l) cosh(gamma l) ]
'''
l = 5.0
z0 = 30.0
alpha = 0.5
beta = 2.0
lossy_media = DefinedGammaZ0(
frequency=Frequency(1, 100, 21, 'GHz'),
gamma=alpha + 1j*beta,
z0=z0
)
ntw = lossy_media.line(d=l, unit='m', z0=z0)
gamma = lossy_media.gamma
npy.testing.assert_array_almost_equal(ntw.a[:,0,0], npy.cosh(gamma*l))
npy.testing.assert_array_almost_equal(ntw.a[:,0,1], z0*npy.sinh(gamma*l))
npy.testing.assert_array_almost_equal(ntw.a[:,1,0], 1.0/z0*npy.sinh(gamma*l))
npy.testing.assert_array_almost_equal(ntw.a[:,1,1], npy.cosh(gamma*l))
def unScaled(): # Set up grid and two-soliton initial data: x = (2.*np.pi/N)*np.arange(-N/2,N/2).reshape(N,1) A, B = 25., 16. A_shift, B_shift = 2., 1. y0 = (3.*A**2.*np.cosh(.5*(A*(x+2.)))**(-2.) + 3*B**2.*np.cosh(.5*(B*(x+1.)))**(-2.)).reshape(N,) k = np.concatenate(( np.arange(0,N/2) , np.array([0]) , np.arange(-N/2+1,0,1) )).reshape(N,) ik3 = 1j*k**3. def F_unscaled(t,u): out = -.5*1j*k*fft(ifft(u,axis=0)**2.,axis=0) + ik3* u return out tmax = .006 dt = .01*N**(-2.) nmax = int(round(tmax/dt)) nplt = int(np.floor((tmax/25.)/dt)) y0 = fft(y0,axis=0) T,Y = RK4(F_unscaled, y0, t0=0, t1=tmax, n=nmax) udata, tdata = np.real(ifft(y0,axis=0)).reshape(N,1), np.array(0.).reshape(1,1) for n in range(1,nmax+1): if np.mod(n,nplt) == 0: t = n*dt u = np.real( ifft(Y[n], axis=0) ).reshape(N,1) udata = np.concatenate((udata,np.nan_to_num(u)),axis=1) tdata = np.concatenate((tdata,np.array(t).reshape(1,1)),axis=1) return x, tdata, udata
def chainPoints(hh, dd, aa, numberOfPoints) :
"""Generate numberOfPoints points (x,y) along a hanging chain from -dd
to dd, given parameter aa. Return lists of x values and y values, each of
length numberOfPoints."""
xs = np.linspace(-dd,dd,numberOfPoints) # [m] x location of points
ys = aa * np.cosh(xs/aa) + (hh - aa * np.cosh(dd / aa)) # [m] y locations
return xs, ys
def height(self,x,t):
xi = x[0] - self.C*(t-1.0) #offsetting start time
eta = self.A1/(np.cosh(self.B*xi))**2 + self.A2/(np.cosh(self.B*xi))**4
# inflowHeightMean already added in wavetank.py
#
# NOTE: x[0] is a vector here for verification only!
return eta
def Kw_gelenk_kik(E, A, I, l, u, w):
"""dyn. Steifigkeit Gelenk an k
:E: @todo
:A: @todo
:I: @todo
:l: @todo
:u: @todo
:w: @todo
:returns: @todo
"""
lam = l * (u * w**2 / E / I) ** (1/4)
eps = l * np.sqrt(u * w**2 / E / A)
o1 = (np.cosh(lam) + np.cos(lam)) / 2
o2 = (np.sinh(lam) + np.sin(lam)) / 2
o3 = (np.cosh(lam) - np.cos(lam)) / 2
o4 = (np.sinh(lam) - np.sin(lam)) / 2
kik = np.matrix([[E*A/l * eps/np.tan(eps), 0, 0],
[0, E*I*lam**3/l**3 * (o1**2 - o2*o4)/(o2*o3 - o1*o4), 0],
[0, 0, 0]])
return kik
def pressure(self,x,t):
""" Gives linearized pressured with P_atm = 0 """
g = (0.0,0.0,-9.81)
z = x[2] - self.h
p = self.rho_0*(-g[2])*self.A* np.cosh(self.kappa*(z+self.h))/np.cosh(self.kappa*self.h) * \
np.exp(1j*(self.k[0]*x[0]+self.k[1]*x[1] - self.omega*t))
return np.real(p)
def velocity_u(self,x,t):
""" Defines a linearized solution to the potential flow
model in two dimensions (x,y,z) for finite depth,
as well as, deep and shllow water limits, for slowly
varying regular wavetrains.
.. todo:: implement deep & shallow water limits. """
theta = self.k[0]*x[0] - self.omega*t # ~ NOTE: x[0] is a vector here!
dtheta = self.diff*theta
z = x[2] - self.h
u = np.zeros(x[0].shape)
# Finite Depth (0 < kh < infty)
for i in range(self.N):
diffPos = (1+self.diff*(i+1))
diffNeg = (1-self.diff*(i+1))
u = u + (-self.g[2]*diffPos*self.k[0]*self.A / (diffPos*self.omega)) * \
np.cosh(diffPos*self.k[0]*(z+self.h))/np.cosh(diffPos*self.k[0]*self.h) * \
np.exp(1j*(i+1)*(theta+(i+1)*dtheta)) + \
(-self.g[2]*diffNeg*self.k[0]*self.A / (diffNeg*self.omega)) * \
np.cosh(diffNeg*self.k[0]*(z+self.h))/np.cosh(diffNeg*self.k[0]*self.h) * \
np.exp(1j*(i+1)*(theta-(i+1)*dtheta))
# Deep water (kh >> 1)
# ... TODO
# Shallow water (kh << 1)
# ... TODO
return np.real(u)
def u2n(n,y,z,lam,gam,beta):
#return (beta*fn(n) + c3(n,lam,gam,beta)*np.cosh(kn(n)/lam*y)+c4(n,lam,gam,beta)*np.sinh(kn(n)/lam*y))*np.cos(kn(n)*z)
top = (beta-beta/np.cosh( kn(n)/lam*gam ) -1.)
bot = (1-beta*np.tanh( -kn(n)/lam )*np.tanh( kn(n)/lam*gam ) )
return ( fn(n)*(beta + (top/bot)*( np.cosh( kn(n)*(y/lam) )*( 1+(1-beta)*(bot/top) ) +
beta*np.tanh( -kn(n)/lam)*np.sinh( kn(n)*(y/lam) ) ) ) )
def grad(data, QofI='U_avg'):
M = data.shape[0]
mu = data[:, 0].reshape((M, 1))
rho = data[:, 1].reshape((M, 1))
dpdx = data[:, 2].reshape((M, 1))
eta = data[:, 3].reshape((M, 1))
B0 = data[:, 4].reshape((M, 1))
Ha = B0 / np.sqrt(eta * mu)
mu0 = 1.0
if (QofI == 'U_avg'):
df_dmu = -dpdx * (np.sqrt(eta * mu) / np.tanh(Ha) -
B0 / np.sinh(Ha)**2) / (2 * B0 * mu**2)
df_drho = np.random.uniform(1.0e-8, 1.0e-10, (M, 1))
df_ddpdx = -(eta * mu - Ha * eta * mu / np.tanh(Ha)) / (mu * B0**2)
df_deta = -dpdx * (2 * eta * mu - Ha * eta * mu / np.tanh(Ha) -
(B0 / np.sinh(Ha))**2) / (2 * eta * mu * B0**2)
df_dB0 = -dpdx * (-2 * eta * mu + Ha * eta * mu / np.tanh(Ha) +
(B0 / np.sinh(Ha))**2) / (mu * B0**3)
elif (QofI == 'B_ind'):
df_dmu = -dpdx * mu0 * (np.sqrt(eta * mu) * np.sinh(Ha) - B0) / (
4 * mu * (B0 * np.cosh(Ha / 2))**2)
df_drho = np.random.uniform(1.0e-8, 1.0e-10, (M, 1))
df_ddpdx = mu0 * (B0 - 2 * np.sqrt(eta * mu) * np.tanh(Ha / 2)) / (
2 * B0**2)
df_deta = -dpdx * mu0 * (np.sqrt(eta * mu) * np.sinh(Ha) - B0) / (
4 * eta * (B0 * np.cosh(Ha / 2))**2)
df_dB0 = -dpdx * mu0 * (B0 + B0 / np.cosh(Ha / 2)**2 - 4 * np.sqrt(
eta * mu) * np.tanh(Ha / 2)) / (2 * B0**3)
return np.concatenate((df_dmu, df_drho, df_ddpdx, df_deta, df_dB0), axis=1)
def CoshXCoshY(data=None, args=None, eqInstance=None, NameOrValueFlag=0):
if NameOrValueFlag: # name used by cache, must be distinct
return sys._getframe().f_code.co_name
try:
return numpy.cosh(data[0]) * numpy.cosh(data[1])
except:
return 1.0E300 * numpy.ones_like(data[0])
def pressure(self,x,t):
""" Gives linearized pressured with P_atm = 0 """
g = (0.0,0.0,-9.81)
z = x[2] - self.h
p = self.rho_0*g[2]*self.A* np.cosh(self.k[0]*(z+self.h))/np.cosh(self.k[0]*self.h) \
* np.exp(1j*self.theta(x,t))
return np.real(p)
def fgrad_y_psi(self, y, psi, return_covar_chain = False):
"""
gradient of f w.r.t to y and psi
returns: NxIx3 tensor of partial derivatives
"""
# 1. exponentiate the a and b (positive!)
mpsi = psi.copy()
w, s, r, d = self.fgrad_y(y, psi, return_precalc = True)
gradients = np.zeros((y.shape[0], y.shape[1], len(mpsi), 3))
for i in range(len(mpsi)):
a,b,c = mpsi[i]
gradients[:,:,i,0] = (b*(1.0/np.cosh(s[i]))**2).T
gradients[:,:,i,1] = a*(d[i] - 2.0*s[i]*r[i]*(1.0/np.cosh(s[i]))**2).T
gradients[:,:,i,2] = (-2.0*a*(b**2)*r[i]*((1.0/np.cosh(s[i]))**2)).T
if return_covar_chain:
covar_grad_chain = np.zeros((y.shape[0], y.shape[1], len(mpsi), 3))
for i in range(len(mpsi)):
a,b,c = mpsi[i]
covar_grad_chain[:, :, i, 0] = (r[i]).T
covar_grad_chain[:, :, i, 1] = (a*(y + c) * ((1.0/np.cosh(s[i]))**2).T)
covar_grad_chain[:, :, i, 2] = a*b*((1.0/np.cosh(s[i]))**2).T
return gradients, covar_grad_chain
return gradients
def fgrad_y_psi(self, y, psi, return_covar_chain = False):
"""
gradient of f w.r.t to y and psi
returns: NxIx4 tensor of partial derivatives
"""
mpsi = psi.copy()
mpsi = mpsi[:self.num_parameters-1].reshape(self.n_terms, 3)
w, s, r, d = self.fgrad_y(y, psi, return_precalc = True)
gradients = np.zeros((y.shape[0], y.shape[1], len(mpsi), 4))
for i in range(len(mpsi)):
a,b,c = mpsi[i]
gradients[:,:,i,0] = (b*(1.0/np.cosh(s[i]))**2).T
gradients[:,:,i,1] = a*(d[i] - 2.0*s[i]*r[i]*(1.0/np.cosh(s[i]))**2).T
gradients[:,:,i,2] = (-2.0*a*(b**2)*r[i]*((1.0/np.cosh(s[i]))**2)).T
gradients[:,:,0,3] = 1.0
if return_covar_chain:
covar_grad_chain = np.zeros((y.shape[0], y.shape[1], len(mpsi), 4))
for i in range(len(mpsi)):
a,b,c = mpsi[i]
covar_grad_chain[:, :, i, 0] = (r[i]).T
covar_grad_chain[:, :, i, 1] = (a*(y + c) * ((1.0/np.cosh(s[i]))**2).T)
covar_grad_chain[:, :, i, 2] = a*b*((1.0/np.cosh(s[i]))**2).T
covar_grad_chain[:, :, 0, 3] = y
return gradients, covar_grad_chain
return gradients
def stretching(sc, Vstretching, theta_s, theta_b):
"""
Computes S-coordinates
INPUT:
sc : normalized levels [ndarray]
Vstretching : ROMS stretching algorithm [int]
theta_s : [int]
theta_b : [int]
hc : [int]
"""
if Vstretching == 1:
# Song and Haidvogel, 1994
cff1 = 1. / np.sinh(theta_s)
cff2 = 0.5 / np.tanh(0.5*theta_s)
C = (1.-theta_b) * cff1 * np.sinh(theta_s * sc) + \
theta_b * (cff2 * np.tanh( theta_s * (sc + 0.5) ) - 0.5)
return C
if Vstretching == 4:
# A. Shchepetkin (UCLA-ROMS, 2010) double vertical stretching function
if theta_s > 0:
Csur = ( 1.0 - np.cosh(theta_s*sc) ) / ( np.cosh(theta_s) -1.0 )
else:
Csur = -sc**2
if theta_b > 0:
Cbot = ( np.exp(theta_b*Csur)-1.0 ) / ( 1.0-np.exp(-theta_b) )
return Cbot
else:
return Csur
def cosh_d(v1, v2):
r1, phi1 = list(v1)[:2]
r2, phi2 = list(v2)[:2]
return max(
1.0, np.cosh(r1) * np.cosh(r2) - np.sinh(r1) * np.sinh(r2) * np.cos(phi1 - phi2)
) # Python precision issues
def transform(self, X, y=None):
"""Apply approximate feature map to X.
Parameters
----------
X: array-like, shape (n_samples, n_features)
Returns
-------
X_new: array-like, shape (n_samples, n_features * (2n + 1))
"""
X = array2d(X)
# check if X has zeros. Doesn't play well with np.log.
if (X <= 0).any():
raise ValueError("Entries of X must be strictly positive.")
X_new = []
# zeroth component
# 1/cosh = sech
X_new.append(np.sqrt(X * self.sample_interval / np.cosh(0)))
log_step = self.sample_interval * np.log(X)
step = 2 * X * self.sample_interval
for j in xrange(1, self.sample_steps):
factor = np.sqrt(step / np.cosh(np.pi * j * self.sample_interval))
X_new.append(factor * np.cos(j * log_step))
X_new.append(factor * np.sin(j * log_step))
return np.hstack(X_new)
def _get_ln_y_ref(self, rup, dists, C):
"""
Get an intensity on a reference soil.
Implements eq. 13a.
"""
# reverse faulting flag
Frv = 1. if 30 <= rup.rake <= 150 else 0.
# normal faulting flag
Fnm = 1. if -120 <= rup.rake <= -60 else 0.
# hanging wall flag
Fhw = np.zeros_like(dists.rx)
idx = np.nonzero(dists.rx >= 0.)
Fhw[idx] = 1.
# a part in eq. 11
mag_test1 = np.cosh(2. * max(rup.mag - 4.5, 0))
# centered DPP
centered_dpp = self._get_centered_cdpp(dists)
# centered_ztor
centered_ztor = self._get_centered_ztor(rup, Frv)
#
ln_y_ref = (
# first part of eq. 11
C['c1']
+ (C['c1a'] + C['c1c'] / mag_test1) * Frv
+ (C['c1b'] + C['c1d'] / mag_test1) * Fnm
+ (C['c7'] + C['c7b'] / mag_test1) * centered_ztor
+ (C['c11'] + C['c11b'] / mag_test1) *
np.cos(math.radians(rup.dip)) ** 2
# second part
+ C['c2'] * (rup.mag - 6)
+ ((C['c2'] - C['c3']) / C['cn'])
* np.log(1 + np.exp(C['cn'] * (C['cm'] - rup.mag)))
# third part
+ C['c4']
* np.log(dists.rrup + C['c5']
* np.cosh(C['c6'] * max(rup.mag - C['chm'], 0)))
+ (C['c4a'] - C['c4'])
* np.log(np.sqrt(dists.rrup ** 2 + C['crb'] ** 2))
# forth part
+ (C['cg1'] + C['cg2'] / (np.cosh(max(rup.mag - C['cg3'], 0))))
* dists.rrup
# fifth part
+ C['c8'] * np.fmax(1 - (np.fmax(dists.rrup - 40,
np.zeros_like(dists)) / 30.),
np.zeros_like(dists))[0]
* min(max(rup.mag - 5.5, 0) / 0.8, 1.0)
* np.exp(-1 * C['c8a'] * (rup.mag - C['c8b']) ** 2) * centered_dpp
# sixth part
+ C['c9'] * Fhw * np.cos(math.radians(rup.dip)) *
(C['c9a'] + (1 - C['c9a']) * np.tanh(dists.rx / C['c9b']))
* (1 - np.sqrt(dists.rjb ** 2 + rup.ztor ** 2)
/ (dists.rrup + 1.0))
)
return ln_y_ref
def regression_test(U, X, N, comm, rank, L, FST, ST, U0, U_hat0,**kw):
Ha = config.Ha
u_exact = ( cosh(Ha) - cosh(Ha*X[0,:,0,0]))/(cosh(Ha) - 1.0)
if rank == 0:
print "Time %2.5f Error %2.12e" %(config.t, linalg.norm(u_exact-U[1,:,0,0],inf))
plt.plot(X[0,:,0,0], U[1,:,0,0], X[0,:,0,0], u_exact,'*r')
plt.show()
def init(U, U_hat, X, FST, ST, SN, t=0.):
Ha = config.Ha
B_strength = config.B_strength
for i in range(U.shape[1]):
x = X[0, i, 0, 0]
for j in range(U.shape[2]):
y = X[1, i, j, 0]
u = 0.
v = (cosh(Ha)-cosh(Ha*x))/(cosh(Ha)-1.0)
Bx = (sinh(Ha*x)-Ha*x*cosh(Ha))/(Ha**2*cosh(Ha))
By = B_strength
U[0, i, j, :] = u
U[1, i, j, :] = v
U[3, i, j, :] = Bx
U[4, i, j, :] = By
U[2] = 0
U[5] = 0
for i in range(6):
if i<3:
U_hat[i] = FST.fst(U[i], U_hat[i], ST)
else:
U_hat[i] = FST.fst(U[i], U_hat[i], SN)
for i in range(6):
if i<3:
U[i] = FST.ifst(U_hat[i], U[i], ST)
else:
U[i] = FST.ifst(U_hat[i], U[i], SN)
for i in range(6):
if i<3:
U_hat[i] = FST.fst(U[i], U_hat[i], ST)
else:
U_hat[i] = FST.fst(U[i], U_hat[i], SN)
def reflection_theory(self, f):
"""
Return the theoretical reflection of the taper.
As a theoreticalk result this reflection doesn't take into account the
impedance of the LJPA.
Parameters
----------
f : float, np.ndarray
Frequency in GHz.
"""
v = (self.beta(f)*self.l)**2. - self.A()**2.
if type(f) is float:
if v < 0:
c = np.cosh(np.sqrt(self.A()**2. - (self.beta(f)*self.l)**2.))
else:
c = np.cos(np.sqrt((self.beta(f)*self.l)**2. - self.A()**2.))
else:
c = np.ones_like(v)
c[v<0] = np.cosh(np.sqrt(self.A()**2. - (self.beta(f[v<0])*self.l)**2.))
c[v>0] = np.cos(np.sqrt((self.beta(f[v>0])*self.l)**2. - self.A()**2.))
return self.gm*np.exp(-1j*self.beta(f)*self.l)*c
def intxfxcoshcurve(x, m):
# measured in x direction (towards point end of curve) from zero
# equation for the integral of xf(x)
intxfx = ( 1/( np.cosh( m*np.pi )-1))*((( np.cosh( m*np.pi )-( m*np.pi )*np.sinh( m*np.pi ))/(( m*np.pi )**2))+(( 1/2 )*np.cosh( m*np.pi ))-( 1/(( m*np.pi )**2 )))
# multiply calculated intxfx by actual dimension in x direction
intxfx = x*intxfx
return intxfx
def process(self, specificLoudness):
# convolution with weighting function
self.smoothSpecificLoudness = np.zeros(specificLoudness.shape)
nFilters = specificLoudness.shape[0]
for ear in range(2):
self.smoothSpecificLoudness[:, ear] = \
np.convolve(
self.fullGaussian, specificLoudness[:, ear]
)[nFilters-1:-nFilters+1]
# Gain computation
self.smoothSpecificLoudness[self.smoothSpecificLoudness < 1e-12] = \
1e-12
inhibLeft = \
2.0 / (1 + (1.0 / np.cosh(
self.smoothSpecificLoudness[:, 1] /
self.smoothSpecificLoudness[:, 0]
)) ** 1.5978)
inhibRight = 2.0 / (1 + (1.0 / np.cosh(
self.smoothSpecificLoudness[:, 0] /
self.smoothSpecificLoudness[:, 1]
)) ** 1.5978)
# Apply gains to original specific loudness
self.inhibitedSpecificLoudness = specificLoudness.copy()
self.inhibitedSpecificLoudness[:, 0] /= inhibLeft
self.inhibitedSpecificLoudness[:, 1] /= inhibRight
def get_ind_potential(mesh,num,amp,omeg,depth,wk):
'''
for cylinder in indefinite water ???
'''
import numpy as np
from dutwav.analytical import bj
from scipy.special import jv
mesh.nodes[1]
g=9.801
res={}
for i in range(num):
x=mesh.nodes[i+1][0]
y=mesh.nodes[i+1][1]
z=mesh.nodes[i+1][2]
theta = np.arctan2(y,x)
r=np.sqrt(x**2+y**2)
tmp=0.
for m in range(10):
eps=2.
if m==0:
eps=1.
tp=-1j*g*amp/omeg
tp=tp*np.cosh(wk*(z+depth))/np.cosh(wk*depth)
tp=tp*eps*(1j)**m*bj(wk*r,m)*np.cos(m*theta)
# tp=tp*eps*(1j)**m*jv(m,wk*r)*np.cos(m*theta)
tmp+=tp
res[i+1]=tmp
return res
def update(rank, X, U, P, N, comm, L, **kw):
Ha = config.Ha
if config.tstep % config.compute_energy == 0:
u_exact = ( cosh(Ha) - cosh(Ha*X[0,:,0,0]))/(cosh(Ha) - 1.0)
if rank == 0:
print "Time %2.5f Error %2.12e" %(config.t, linalg.norm(u_exact-U[1,:,0,0],inf))
def N2sol(t,z,b2,gamma, t0):
"""
complete field for the N=2 soliton
"""
p01 = np.abs(b2) / gamma / t0**2
ld = t0**2 / np.abs(b2)
return np.sqrt(p01) * 4 * (np.cosh(3*t/t0) + 3 * np.exp(1.0j * 4 * z/ld ) * np.cosh(t/t0)) / (np.cosh(4*t/t0)+4*np.cosh(2*t/t0)+ 3*np.cos(4*z/ld)) * np.exp( 1.0j * z/ld /2.)
def bipartivity_exact(G, focal=None):
G_MAT=NX.to_numpy_matrix(G) # Convert NX network to adjacency matrix
ei,ev=N.linalg.eig(G_MAT) # Calculate eigenvalues abd eigenvectors
SC_even=0 # Sum of the contributions from even closed walks in G
SC_all=0 # Sum of the contributions of all closed walks in G
if focal is None:
''' Formulas described on page 2
First code block calculates bipartivity of G globally'''
for j in range(0,G.number_of_nodes()):
SC_even=SC_even+N.cosh(ei[j].real)
SC_all=SC_all+N.e**(ei[j].real)
# Proportion of even closed walks over all closed walks
B=SC_even/SC_all
else:
# If focal node is passed as str, or object
if focal is not int:
n=G.nodes()
focal=n.index(focal)
'''Second code block calculates contibution of 'focal' to bipartivity'''
for j in range(0,G.number_of_nodes()):
SC_even=SC_even+((ev[focal,j].real)**2)*(N.cosh(ei[j].real))
SC_all=SC_all+((ev[focal,j].real)**2)*(N.e**(ei[j].real))
B=SC_even/SC_all # Proportion of even CW for focal node, ie how much
# the focal node contributes to bipartivity.
return B
def hyperbola(self, p1, p2, toa1, toa2):
'''
calculates the asymptotes of hyperbola that represents equidifference
curves for time of arrival (TDOA) for two points.
input:
p1,p2 location in the form (lat,lon)
toa1,toa2 timestamp in seconds
return:
(x1,y1,x2,y2) x & y coordinates of asymptotes 1 & 2
'''
if DEBUG:
print "p1: ", p1
print "p2: ", p2
print "toa1: ", repr(toa1)
print "toa2: ", repr(toa2)
print "type(toa1): ", type(toa1)
print "type(toa2): ", type(toa2)
x1 = p1[0]
y1 = p1[1]
x2 = p2[0]
y2 = p2[1]
# angle between two locations and 'horizontal'
alpha = np.arctan((y2 - y1) / (x2 - x1))
if DEBUG:
print "alpha: ", alpha
# approx. midpoint between two locations
midpoint = self.midpoint(p1, p2)
x_0 = midpoint[0]
y_0 = midpoint[1]
# 3) determine hyperbola asymptotes
a = (0.5) * self.speed_of_light * (np.abs(toa1 - toa2)) / (60 * 1852)
distance = self.distance(p1, p2) / (60 * 1852)
b = np.sqrt((distance / 2.0)**2 - a**2)
asymp_slope = (b / a, -b / a)
# hyperbolas
t = np.linspace(-np.pi, np.pi, 100)
print 'type(t): ', type(t)
# t = np.linspace(-2,2,100)
dd = self.speed_of_light * ((toa1 - toa2)) / (60 * 1852)
if x1 < x2:
if DEBUG:
print "x1 < x2"
print "dd: ", dd
if dd < 0:
if DEBUG:
print "dd < 0"
x_hyperbola = -a * np.cosh(t)
y_hyperbola = b * np.sinh(t)
else: # dd > 0
if DEBUG:
print "dd > 0"
x_hyperbola = a * np.cosh(t)
y_hyperbola = b * np.sinh(t)
else: # x1 > x2
if DEBUG:
print "x1 > x2"
print "dd: ", dd
if dd > 0:
if DEBUG:
print "dd > 0"
x_hyperbola = -a * np.cosh(t)
y_hyperbola = b * np.sinh(t)
else: # dd < 0:
if DEBUG:
print "dd < 0"
x_hyperbola = a * np.cosh(t)
y_hyperbola = b * np.sinh(t)
x_hyperbola_p = x_hyperbola * np.cos(alpha) - y_hyperbola * np.sin(
alpha) + x_0
y_hyperbola_p = x_hyperbola * np.sin(alpha) + y_hyperbola * np.cos(
alpha) + y_0
#plt.plot(x_hyperbola_p,y_hyperbola_p)
return (x_hyperbola_p, y_hyperbola_p)
import numpy as np
import matplotlib.pyplot as plt
import math
import time
import argparse
sigmoid = (lambda x: 1. / (1 + np.exp(-x)), lambda x: x * (1. - x))
relu = (lambda x: np.maximum(0, x), lambda x: 1. * (x > 0))
tanh = (lambda x: np.tanh(x), lambda x: (1. / np.cosh(x))**2)
class NeuralNetwork:
def __init__(self, x, y, activation1, activation2, eta=0.1, layer_size=4):
self.input = x
input_size = self.input.shape[1]
self.weights1 = np.random.rand(layer_size, input_size)
self.weights2 = np.random.rand(1, layer_size)
self.y = y
self.output = np.zeros(self.y.shape)
self.eta = eta
self.activation1, self.activation_derivative1 = activation1
self.activation2, self.activation_derivative2 = activation2
def feedforward(self):
self.layer1 = self.activation1(np.dot(self.input, self.weights1.T))
self.output = self.activation2(np.dot(self.layer1, self.weights2.T))
def backprop(self):
delta2 = (self.y - self.output) * \
def sech(x):
"""Hyperbolic secant"""
return 1 / np.cosh(x)
def sechsoln(xspace, t, kappa):
return 1 / np.sqrt(2) * kappa * 1 / np.cosh(kappa * xspace) * np.exp(
1j * kappa**2 * t)
def euler_beam_modes(n=10, bctype=3, npoints=2001,
beamparams=np.array([7.31e10, 8.4375e-09,
2747.0, 4.5e-04, 0.4])):
"""Mode shapes and natural frequencies of Euler-Bernoulli beam.
Parameters
----------
n: int, numpy array
highest mode number or array of mode numbers to return
bctype: int
bctype = 1 free-free
bctype = 2 clamped-free
bctype = 3 clamped-pinned
bctype = 4 clamped-sliding
bctype = 5 clamped-clamped
bctype = 6 pinned-pinned
beamparams: numpy array
E, I, rho, A, L,
Young's modulus, second moment of area, density, cross section area,
length of beam
npoints: int
number of points for returned mode shape array
Returns
-------
omega_n: numpy array
array of natural frequencies
x: numpy array
x coordinate
U: numpy array
mass normalized mode shape
Examples
--------
>>> import matplotlib.pyplot as plt
>>> import vibration_toolbox as vtb
>>> omega_n, x, U = vtb.euler_beam_modes(n=1)
>>> plt.figure()
<matplotlib.figure...>
>>> plt.plot(x,U)
[<matplotlib.lines.Line2D object at ...>]
>>> plt.xlabel('x (m)')
<matplotlib.text.Text object at ...>
>>> plt.ylabel('Displacement (m)')
<matplotlib.text.Text object at ...>
>>> plt.title('Mode 1')
<matplotlib.text.Text object at ...>
>>> plt.grid('on')
"""
E = beamparams[0]
I = beamparams[1]
rho = beamparams[2]
A = beamparams[3]
L = beamparams[4]
if isinstance(n, int):
ln = n
n = np.arange(n) + 1
else:
ln = len(n)
# len=[0:(1/(npoints-1)):1]'; %Normalized length of the beam
x_normed = np.linspace(0, 1, npoints, endpoint=True)
x = x_normed * L
# Determine natural frequencies and mode shapes depending on the
# boundary condition.
# Mass simplification. The following was arange_(1,length_(n)).reshape(-1)
mode_num_range = np.arange(0, ln)
Bnl = np.empty(ln)
w = np.empty(ln)
U = np.empty([npoints, ln])
if bctype == 1:
desc = 'Free-Free '
Bnllow = np.array((0, 0, 4.73004074486, 7.8532046241,
10.995607838, 14.1371654913, 17.2787596574))
for i in mode_num_range:
if n[i] > 7:
Bnl[i] = (2 * n[i] - 3) * np.pi / 2
else:
Bnl[i] = Bnllow[i]
for i in mode_num_range:
if n[i] == 1:
w[i] = 0
U[:, i] = 1 + x_normed * 0
elif n[i] == 2:
w[i] = 0
U[:, i] = x_normed - 0.5
else:
sig = (np.cosh(Bnl[i]) - np.cos(Bnl[i])) / \
(np.sinh(Bnl[i]) - np.sin(Bnl[i]))
w[i] = (Bnl[i] ** 2) * np.sqrt(E * I / (rho * A * L ** 4))
b = Bnl[i] * x_normed
U[:, i] = np.cosh(b) + np.cos(b) - sig * \
(np.sinh(b) + np.sin(b))
elif bctype == 2:
desc = 'Clamped-Free '
Bnllow = np.array((1.88, 4.69, 7.85, 10.99, 14.14))
for i in mode_num_range:
if n[i] > 4:
Bnl[i] = (2 * n[i] - 1) * np.pi / 2
else:
Bnl[i] = Bnllow[i]
for i in mode_num_range:
sig = (np.sinh(Bnl[i]) - np.sin(Bnl[i])) / \
(np.cosh(Bnl[i]) - np.cos(Bnl[i]))
w[i] = (Bnl[i] ** 2) * np.sqrt(E * I / (rho * A * L ** 4))
b = Bnl[i] * x_normed
# plt.plot(x,(sp.cosh(b) - sp.cos(b) -
# sig * (sp.sinh(b) - sp.sin(b))))
U[:, i] = np.cosh(b) - np.cos(b) - sig * (np.sinh(b) - np.sin(b))
elif bctype == 3:
desc = 'Clamped-Pinned '
Bnllow = np.array((3.93, 7.07, 10.21, 13.35, 16.49))
for i in mode_num_range:
if n[i] > 4:
Bnl[i] = (4 * n[i] + 1) * np.pi / 4
else:
Bnl[i] = Bnllow[i]
for i in mode_num_range:
sig = (np.cosh(Bnl[i]) - np.cos(Bnl[i])) / \
(np.sinh(Bnl[i]) - np.sin(Bnl[i]))
w[i] = (Bnl[i] ** 2) * np.sqrt(E * I / (rho * A * L ** 4))
b = Bnl[i] * x_normed
U[:, i] = np.cosh(b) - np.cos(b) - sig * (np.sinh(b) - np.sin(b))
elif bctype == 4:
desc = 'Clamped-Sliding '
Bnllow = np.array((2.37, 5.5, 8.64, 11.78, 14.92))
for i in mode_num_range:
if n[i] > 4:
Bnl[i] = (4 * n[i] - 1) * np.pi / 4
else:
Bnl[i] = Bnllow[i]
for i in mode_num_range:
sig = (np.sinh(Bnl[i]) + np.sin(Bnl[i])) / \
(np.cosh(Bnl[i]) - np.cos(Bnl[i]))
w[i] = (Bnl[i] ** 2) * np.sqrt(E * I / (rho * A * L ** 4))
b = Bnl[i] * x_normed
U[:, i] = np.cosh(b) - np.cos(b) - sig * (np.sinh(b) - np.sin(b))
elif bctype == 5:
desc = 'Clamped-Clamped '
Bnllow = np.array((4.73, 7.85, 11, 14.14, 17.28))
for i in mode_num_range:
if n[i] > 4:
Bnl[i] = (2 * n[i] + 1) * np.pi / 2
else:
Bnl[i] = Bnllow[i]
for i in mode_num_range:
sig = (np.cosh(Bnl[i]) - np.cos(Bnl[i])) / \
(np.sinh(Bnl[i]) - np.sin(Bnl[i]))
w[i] = (Bnl[i] ** 2) * np.sqrt(E * I / (rho * A * L ** 4))
b = Bnl[i] * x_normed
U[:, i] = np.cosh(b) - np.cos(b) - sig * (np.sinh(b) - np.sin(b))
elif bctype == 6:
desc = 'Pinned-Pinned '
for i in mode_num_range:
Bnl[i] = n[i] * np.pi
w[i] = (Bnl[i] ** 2) * np.sqrt(E * I / (rho * A * L ** 4))
U[:, i] = np.sin(Bnl[i] * x_normed)
# Mass Normalization of mode shapes
for i in mode_num_range:
U[:, i] = U[:, i] / np.sqrt(np.dot(U[:, i], U[:, i]) * rho * A * L)
omega_n = w
return omega_n, x, U
def coth(x):
return np.cosh(x) / np.sinh(x)
def d_tanh(data):
return 1 / (np.cosh(data))**2
def kernel(k, a=a, b=b):
if k:
return sinh(a * k) / cosh(b * k)
return 0
if 25 in pids:
#print "Higgs"
for pin in range(0, branchParticle.GetEntriesFast()):
#for pin in range(0, 20):
part = branchParticle.At(pin)
if (abs(part.PID) == 24 and part.Status == 22):
wpart.SetPtEtaPhiE(part.PT, part.Eta, part.Phi, part.E)
# No higgs event
else:
print "No Higgs!"
for etrenrty in range(0, branchTrack.GetEntriesFast()):
eterm = branchTrack.At(etrenrty)
temp.SetPtEtaPhiE(eterm.PT, eterm.Eta, eterm.Phi,
eterm.PT * np.cosh(eterm.Eta))
vectors.append((temp.E(), temp.Px(), temp.Py(), temp.Pz()))
for pentry in range(0, branchPhoton.GetEntriesFast()):
pterm = branchPhoton.At(pentry)
temp.SetPtEtaPhiE(pterm.ET, pterm.Eta, pterm.Phi, pterm.E)
vectors.append((temp.E(), temp.Px(), temp.Py(), temp.Pz()))
for nentry in range(0, branchNeutral.GetEntriesFast()):
nterm = branchNeutral.At(nentry)
temp.SetPtEtaPhiE(nterm.ET, nterm.Eta, nterm.Phi, nterm.E)
vectors.append((temp.E(), temp.Px(), temp.Py(), temp.Pz()))
vects = np.asarray(vectors,
dtype=np.dtype([('E', 'f8'), ('px', 'f8'), ('py', 'f8'),
('pz', 'f8')]))
def kernel(k, a=a, b=b):
return cosh(a * k) / cosh(b * k)
def tanh_fn(x): #domain -1 - 1 return np.sinh( x ) / np.cosh( x )
def BEMT(J, P_D, BA_ratio, N_blades, chord, relative_pitch_distribution,
lift_curve_slope, root_thickness):
'''
Main Blade Element Momentum Theory Function. Uses algorithm form referenced
book.
Parameters
----------
J: float
advance ratio
P_D: float
pitch diameter ratio at 0.7 radius
BA_ratio: float
blade area ratio
N_blades: int
number of blades
chord: array of floats
chord of blade at different radial positions
relative_pitch_distribution: array of float
pitch distribution at radial positions relative to pitch distibution at
x_R = 0.7
lift_curve_slope: float
lift curve slope of blade
root_thickness: float
thickness at blade root
Returns
-------
Total_Thrust: float
total thrust coefficient along blade
Total_Torque: float
total torque coefficient along blade
Open_eta: float
open water efficiency
'''
assert J <= 1.1 and J >= 0.3, ' Advance ratio must be between 0.3 and 1.1 \
but has value {}'.format(J)
assert P_D >= J + 0.03 and P_D >= 0.6 and P_D <= 1.2, 'Pitch/Diameter Ratio at \
0.7R is outside limits for this J value'
assert BA_ratio >= 0.4 and BA_ratio <= 0.8, ' Blade area ratio is out limits.\
Must be between 0.4 and 0.8 but has value {}'.format(BA_ratio)
assert N_blades >= 3 and N_blades <= 5, 'Number of blades must be between 3 \
and 5 but is of value {}'.format(N_blades)
assert type(
J
) == np.float or np.float64, 'J is not of type float but is of type {}'.format(
type(J))
assert type(
P_D
) == np.float or np.float64, 'P_D is not of type float but is of type {}'.format(
type(P_D))
assert type(
BA_ratio
) == np.float or np.float64, 'BA_ratio is not of type float but is of type {}'.format(
type(BA_ratio))
assert type(
N_blades
) == int, 'N_blades is not of type int but is of type {}'.format(
type(N_blades))
assert type(
chord
) == np.array or np.ndarray, 'chord is not of type array but is of type {}'.format(
type(chord))
assert type(
relative_pitch_distribution
) == np.array or np.ndarray, 'relative_pitch_distribution is not of type array but is of type {}'.format(
type(relative_pitch_distribution))
assert type(
lift_curve_slope
) == np.float or np.float64, 'lift_curve_slope is not of type float but is of type {}'.format(
type(lift_curve_slope))
assert type(
root_thickness
) == np.float or np.float64, 'root_thickness is not of type float but is of type {}'.format(
type(root_thickness))
# create arrays
x_R = np.zeros_like(chord)
phi_plus_alhpa = np.zeros_like(chord)
tan_psi = np.zeros_like(chord)
local_P_D = np.zeros_like(chord)
K_series = np.zeros_like(chord)
chord_diameter = np.zeros_like(chord)
thickness_chord = np.zeros_like(chord)
KT_dx = np.zeros_like(chord)
Cl = np.zeros_like(chord)
Cd = np.zeros_like(chord)
KQ_dx = np.zeros_like(chord)
phi = np.zeros_like(chord)
alpha = np.zeros_like(chord)
K2 = np.zeros_like(chord)
MC = np.zeros_like(chord)
MT = np.zeros_like(chord)
CC = np.zeros_like(chord)
LI = 0.8 * 12.67
LM = 0.8 * 11.3
VV = 0
for i in range(1, 9):
# calculate local pitch
local_P_D[i] = P_D * relative_pitch_distribution[i]
# set value of x_R for each element
x_R[i] = (i + 1) / 10
# compute chord/diameter from blade geometry
chord_diameter[i] = chord[i] * BA_ratio * 4.0 / (N_blades * 0.5)
# compute Nc/D for later use
Nc_D = N_blades * chord_diameter[i]
# thickness distribution from rootthickness
thickness_distribution = root_thickness - (root_thickness *
0.935) * x_R[i]
# thickness/chord
thickness_chord[i] = thickness_distribution / chord_diameter[i]
# set initial angle of attack to 0
alpha[i] = 0.0
# set number of alpha iterations to 0
alpha_iterations = 0
# set convergence flag to 0
alpha_converge = 0
# start of alpha convergenc loop
while alpha_iterations < 2000 and alpha_converge == 0:
alpha_iterations += 1
# calculate tan of inflow angle
tan_psi[i] = J / (np.pi * x_R[i])
# induced flow angle plus angle of attack
phi_plus_alhpa[i] = np.arctan2(local_P_D[i], (np.pi * x_R[i]))
# inflow angle
phi[i] = phi_plus_alhpa[i] - alpha[i]
# ideal efficieny
eta_ideal = tan_psi[i] / np.tan(phi[i])
# initially set efficiency to ideal
eta = 0.9 * eta_ideal
gamma = 0.0 # zero for ideal efficiency
KF = np.tan(phi[i]) * x_R[i]
# Goldstein Correction
SF = N_blades / (2.0 * np.tan(phi[i]) * x_R[i]) - 0.5
F1 = np.cosh(SF)
F2 = np.cosh(SF * x_R[i])
F3 = F2 / F1
F4 = np.arccos(F3)
K = 2.0 * F4 / np.pi
# store factor in array
K_series[i] = K
# set number of efficiency iterations to 0
eta_iterations = 0
eta_converge = 0
while eta_iterations < 2000 and eta_converge == 0:
# axial inflow factor
a = (1 - eta_ideal) / (eta_ideal + (tan_psi[i]**2) / eta)
# local thrust coefficient
KT_dx[i] = np.pi * (J**2) * x_R[i] * K * a * (1 + a)
# circumferential inflow factor (a')
a_p = 1 - eta_ideal * (1 + a)
# lift coefficient
Cl[i] = KT_dx[i]/(((np.pi**2)/4)*(N_blades*chord_diameter[i])*(x_R[i]**2)*((1-a_p)**2) \
* (1/np.cos(phi[i]))*(1-np.tan(phi[i])*np.tan(gamma)))
# compute drag coefficient from angle of attack and geometry
F6 = 0.0107 + (alpha[i] + 1.0) * (-0.0015 + alpha[i] *
(0.0015 + 0.000965 *
(alpha[i] - 1.0)))
F7 = -0.03833 + (alpha[i] + 1.0) * (0.0133 + alpha[i] *
(-0.015 - 0.01166 *
(alpha[i] - 1.0)))
F8 = 0.8193 + (alpha[i] + 1.0) * (-0.0138 + alpha[i] *
(0.0903 + 0.079 *
(alpha[i] - 1.0)))
F9 = -3.076 + (alpha[i] + 1.0) * (-0.0728 + alpha[i] *
(-0.3162 - 0.2437 *
(alpha[i] - 1.0)))
Cd[i] = F6 + thickness_chord[i] * (
F7 + (thickness_chord[i] - 0.06) *
(F8 + F9 * (thickness_chord[i] - 0.12)))
# update gamma
gamma = np.arctan2(Cd[i], Cl[i])
# compute efficieny
eta_calculated = tan_psi[i] / (np.tan((phi[i] + gamma)))
eta_iterations += 1
if np.abs(eta_calculated - eta) < 0.001:
# if old efficieny = new efficiency
# set convergence flag to 1 to stop iteration
eta_converge = 1
# camber correction
# geometric
K2[1] = 1.0 + 2.857 * (BA_ratio - 0.4) * (
BA_ratio - 0.6) * (BA_ratio - 0.9)
K2[2] = 1.19 + (BA_ratio - 0.4) * (BA_ratio - 0.6) * (
0.267 + (BA_ratio - 0.9) * 0.1665)
K2[3] = 1.3 + (BA_ratio -
0.4) * (0.1 + (BA_ratio - 0.6) *
(1 + (BA_ratio - 0.9) * 0.43))
K2[4] = 1.54 + (BA_ratio - 0.4) * (0.15 +
(BA_ratio - 0.6) *
(1.1 + 0.286 *
(BA_ratio - 0.9)))
K2[5] = 1.67 + (BA_ratio -
0.4) * (0.55 + (BA_ratio - 0.6) *
(0.1667 +
(BA_ratio - 0.9) * 2.9465))
K2[6] = 1.8 + (BA_ratio -
0.4) * (0.75 + (BA_ratio - 0.6) *
(0.3 + (BA_ratio - 0.9) * 2.835))
K2[7] = 1.8 + (BA_ratio -
0.4) * (1.0 + (BA_ratio - 0.6) *
(1.333 + (BA_ratio - 0.9) * 1.905))
K2[8] = 1.75 + (BA_ratio -
0.4) * (1.25 + (BA_ratio - 0.6) *
(1.5 + (BA_ratio - 0.9) * 3.55))
# from camber correction curves
# KF is lambda from goldstein correction
U1 = -0.65 * KF * KF + 1.1 * KF + 0.664
U2 = (0.85 + (KF - 0.3) * (-4.0 + (KF - 0.4) *
(15.42 - 47.95 * (KF - 0.5))))
U2 = -0.09 + (KF - 0.2) * U2
U3 = (1.375 + (KF - 0.3) * (-3.75 + (KF - 0.4) *
(20.85 - 75.7875 *
(KF - 0.5))))
U3 = -0.2 + (KF - 0.2) * U3
K1 = U1 + (BA_ratio - 0.4) * (U2 + U3 * (BA_ratio - 0.8))
# camber correction
CC[i] = K1 * K2[i]
# compute angle of attack using camber correction
# MC = camber/chord
# MT = camber/thickness
# multiply by pi/180 as lift curve slope is degrees
if VV == 1:
MC[i] = MT[i] * thickness_chord[i]
alpha_calculated = (CC[i] * Cl[i] / LI -
MC[i]) * LM / 0.1097 + (
MC[i] * LI / lift_curve_slope)
elif VV == 5:
MC[i] = 0.5 * thickness_chord[i]
alpha_calculated = (CC[i] * Cl[i] / LI -
MC[i]) * LM / 0.1097 + (
Cl[i] / lift_curve_slope)
alpha_calculated = alpha_calculated * np.pi / 180
MT[i] = MC[i] / thickness_chord[i]
else:
alpha_calculated = Cl[i] / lift_curve_slope
alpha_calculated = alpha_calculated * np.pi / 180
MC[i] = Cl[i] / LI
MC[i] = MC[i] * CC[i]
if MC[i] < (0.5 * thickness_chord[i]):
MT[i] = MC[i] / thickness_chord[i]
else:
VV = 5
MC[i] = 0.5 * thickness_chord[i]
alpha_calculated = (CC[i] * Cl[i] / LI -
MC[i]) * LM / 0.1097 + (
Cl[i] / lift_curve_slope)
alpha_calculated = alpha_calculated * np.pi / 180
MT[i] = MC[i] / thickness_chord[i]
else:
# set the new efficiency to calculated efficiency
eta = eta_calculated
if np.abs((alpha_calculated - alpha[i]) / alpha_calculated) < 0.1:
# compute local torque coefficient
KQ_dx[i] = 4.935 * J * (x_R[i]**3) * K * a_p * (1 + a)
# set convergence flag to 1 to stop iteration
alpha_converge = 1
else:
alpha[i] = (alpha_calculated + alpha[i]) / 2.0
# set tip coefficients to 0
x_R[9] = 1.0
KT_dx[9] = 0.0
KQ_dx[9] = 0.0
K_series[9] = 0.0
local_P_D[9] = local_P_D[8]
alpha[9] = alpha[8]
print(KT_dx)
# Integrate over blade to compute total coefficients. Using simpsons rule
Total_Thrust = (KT_dx[1] + 2.0 * (KT_dx[3] + KT_dx[5] + KT_dx[7]) + 4.0 *
(KT_dx[2] + KT_dx[4] + KT_dx[6] + KT_dx[8]))
Total_Thrust = 0.1 * Total_Thrust / 3.0
Total_Torque = (KQ_dx[1] + 2.0 * (KQ_dx[3] + KQ_dx[5] + KQ_dx[7]) + 4.0 *
(KQ_dx[2] + KQ_dx[4] + KQ_dx[6] + KQ_dx[8]))
Total_Torque = (0.1 * Total_Torque) / 3.0
# open water efficiency
Open_eta = (J * Total_Thrust) / (2.0 * 3.1416 * Total_Torque)
return Total_Thrust, Total_Torque, Open_eta
def g_derivative(self, x):
cosh2 = (np.cosh(self.beta * x))**2
return self.beta / cosh2
def select_events(events, apply_exclusive=True):
selections_ = []
counts_ = []
selections_.append("All")
counts_.append(len(events))
msk_2muons = (events.nMuonCand >= 2)
events_2muons = events[msk_2muons]
dphi = events_2muons.MuonCand.phi[:, 0] - events_2muons.MuonCand.phi[:, 1]
dphi = np.where(dphi >= scipy.constants.pi, dphi - 2 * scipy.constants.pi,
dphi)
dphi = np.where(dphi < -scipy.constants.pi, dphi + 2 * scipy.constants.pi,
dphi)
acopl = 1. - np.abs(dphi) / scipy.constants.pi
events_2muons["Acopl"] = acopl
m1 = events_2muons.MuonCand[:, 0]
m2 = events_2muons.MuonCand[:, 1]
invariant_mass = np.sqrt(
2 * m1.pt * m2.pt *
(np.cosh(m1.eta - m2.eta) - np.cos(m1.phi - m2.phi)))
events_2muons["InvMass"] = invariant_mass
energy_com = 13000.
xi_mumu_plus = (1. / energy_com) * (m1.pt * np.exp(m1.eta) +
m2.pt * np.exp(m2.eta))
xi_mumu_minus = (1. / energy_com) * (m1.pt * np.exp(-m1.eta) +
m2.pt * np.exp(-m2.eta))
events_2muons["XiMuMuPlus"] = xi_mumu_plus
events_2muons["XiMuMuMinus"] = xi_mumu_minus
pfCands_ = events_2muons.PfCand
pfCands_["dR_0"] = np.sqrt(
(pfCands_.eta - events_2muons.MuonCand.eta[:, 0])**2 +
(pfCands_.phi - events_2muons.MuonCand.phi[:, 0])**2)
pfCands_["dR_1"] = np.sqrt(
(pfCands_.eta - events_2muons.MuonCand.eta[:, 1])**2 +
(pfCands_.phi - events_2muons.MuonCand.phi[:, 1])**2)
pfCands_sel1_ = pfCands_[pfCands_.fromPV == 3.0]
pfCands_sel2_ = pfCands_sel1_[pfCands_sel1_.dR_0 > 0.3]
pfCands_sel3_ = pfCands_sel2_[pfCands_sel2_.dR_1 > 0.3]
events_2muons["nExtraPfCandPV3"] = ak.num(pfCands_sel3_)
msk_muon = (np.array(events_2muons.MuonCand.pt[:, 0] >= 50.)
& np.array(events_2muons.MuonCand.pt[:, 1] >= 50.)
& np.array(events_2muons.MuonCand.istight[:, 0] == 1)
& np.array(events_2muons.MuonCand.istight[:, 1] == 1)
& np.array((events_2muons.MuonCand.charge[:, 0] *
events_2muons.MuonCand.charge[:, 1]) == -1))
selections_.append("Muon")
counts_.append(np.sum(msk_muon))
msk_vtx = msk_muon & (np.array(
np.abs(events_2muons.PrimVertexCand.z[:, 0]) <= 15.) & np.array(
np.abs(events_2muons.MuonCand.vtxz[:, 0] -
events_2muons.PrimVertexCand.z[:, 0]) <= 0.02) & np.array(
np.abs(events_2muons.MuonCand.vtxz[:, 1] -
events_2muons.PrimVertexCand.z[:, 0]) <= 0.02))
selections_.append("Vertex")
counts_.append(np.sum(msk_vtx))
events_sel = None
if apply_exclusive:
msk_excl = msk_vtx & (np.array(events_2muons["InvMass"] >= 110.)
& np.array(events_2muons["Acopl"] <= 0.009) &
np.array(events_2muons["nExtraPfCandPV3"] <= 1))
selections_.append("Exclusive")
counts_.append(np.sum(msk_excl))
events_sel = events_2muons[msk_excl]
else:
events_sel = events_2muons[msk_vtx]
selections_ = np.array(selections_)
counts_ = np.array(counts_)
return (events_sel, selections_, counts_)
def np_func(x, y):
return np.sinh(x) + np.cosh(y)
x,y = x_split[j],y_split[j]
# initial guesses
x_tail,y_tail = x[1:treshold], y[1:treshold]
slope, intercept = np.polyfit(x_tail,y_tail,1)
p0 = [max(y),0.1,x[np.argmin(np.abs(x))],slope]
lower = [0,0.01,-100,0.9*p0[3]]
upper = [1.1*p0[0],100,100,1.1*p0[3]]
popt, pcov = curve_fit(hysteresis,x,y,p0=p0,maxfev=10000,bounds=[lower,upper])
fit_params.append(popt),fit_errors.append(np.diag(pcov))
# error calculation
dA = lambda x: np.tanh(popt[1]*(x-popt[2]))
dB = lambda x: popt[0]*(x-popt[2])/np.cosh(popt[1]*(x-popt[2]))**2
dC = lambda x: -popt[0]*popt[1]/np.cosh(popt[1]*(x-popt[2]))**2
dD = lambda x: x
grad, dH = np.array([[dA(x),dB(x),dC(x),dD(x)] for x in X]), np.zeros_like(X)
for k,x_val in enumerate(X):
dH[k] += np.sqrt(grad[k].T @ pcov @ grad[k])
A, A_err = 0.5*(fit_params[0][0]+fit_params[1][0]), 0.5*np.sqrt(fit_errors[0][0]**2+fit_errors[1][0]**2)
B, B_err = 0.5*(fit_params[0][1]+fit_params[1][1]), 0.5*np.sqrt(fit_errors[0][1]**2+fit_errors[1][1]**2)
C, C_err = 0.5*(fit_params[0][2]-fit_params[1][2]), 0.5*np.sqrt(fit_errors[0][2]**2+fit_errors[1][2]**2)
D, D_err = 0.5*(fit_params[0][3]+fit_params[1][3]), 0.5*np.sqrt(fit_errors[0][3]**2+fit_errors[1][3]**2)
print(f"SI {temperature} degreecelsius & SI {A:.4f}pm{A_err:.4f} millidegree & SI {B*1e3:.4f}pm{B_err*1e3:.4f} permillioersted \
& SI {C:.2f}pm{C_err:.2f} oersted & SI {D*1e3:.2f}pm{D_err*1e3:.2f} microdegreeperoersted")
def derivative_activation(self, z):
return (np.cosh(z))**(-2)
def _stats(self, lambda_):
mu = 1 / expm1(lambda_)
var = exp(-lambda_) / (expm1(-lambda_))**2
g1 = 2 * cosh(lambda_ / 2.0)
g2 = 4 + 2 * cosh(lambda_)
return mu, var, g1, g2
def w2k(w, theta=0.0, h=inf, g=9.81, count_limit=100):
'''
Translates from frequency to wave number
using the dispersion relation
Parameters
----------
w : array-like
angular frequency [rad/s].
theta : array-like, optional
direction [rad].
h : real scalar, optional
water depth [m].
g : real scalar or array-like of size 2.
constant of gravity [m/s**2] or 3D normalizing constant
Returns
-------
k1, k2 : ndarray
wave numbers [rad/m] along dimension 1 and 2.
Description
-----------
Uses Newton Raphson method to find the wave number k in the dispersion
relation
w**2= g*k*tanh(k*h).
The solution k(w) => k1 = k(w)*cos(theta)
k2 = k(w)*sin(theta)
The size of k1,k2 is the common shape of w and theta according to numpy
broadcasting rules. If w or theta is scalar it functions as a constant
matrix of the same shape as the other.
Example
-------
>>> import pylab as plb
>>> import wafo.wave_theory.dispersion_relation as wsd
>>> w = plb.linspace(0,3);
>>> h = plb.plot(w,w2k(w)[0])
>>> wsd.w2k(range(4))[0]
array([ 0. , 0.1019368 , 0.4077472 , 0.91743119])
>>> wsd.w2k(range(4),h=20)[0]
array([ 0. , 0.10503601, 0.40774726, 0.91743119])
>>> plb.close('all')
See also
--------
k2w
'''
wi, th, hi, gi = atleast_1d(w, theta, h, g)
if wi.size == 0:
return zeros_like(wi)
k = 1.0 * sign(wi) * wi**2.0 / gi[0] # deep water
if (hi > 10.**25).all():
k2 = k * sin(th) * gi[0] / gi[-1] # size np x nf
k1 = k * cos(th)
return k1, k2
if gi.size > 1:
raise ValueError('Finite depth in combination with 3D normalization' +
' (len(g)=2) is not implemented yet.')
find = flatnonzero
eps = finfo(float).eps
oshape = k.shape
wi, k, hi = wi.ravel(), k.ravel(), hi.ravel()
# Newton's Method
# Permit no more than count_limit iterations.
hi = hi * ones_like(k)
hn = zeros_like(k)
ix = find((wi < 0) | (0 < wi))
# Break out of the iteration loop for three reasons:
# 1) the last update is very small (compared to x)
# 2) the last update is very small (compared to sqrt(eps))
# 3) There are more than 100 iterations. This should NEVER happen.
count = 0
while (ix.size > 0 and count < count_limit):
ki = k[ix]
kh = ki * hi[ix]
hn[ix] = (ki * tanh(kh) - wi[ix] ** 2.0 / gi) / \
(tanh(kh) + kh / (cosh(kh) ** 2.0))
knew = ki - hn[ix]
# Make sure that the current guess is not zero.
# When Newton's Method suggests steps that lead to zero guesses
# take a step 9/10ths of the way to zero:
ksmall = find(abs(knew) == 0)
if ksmall.size > 0:
knew[ksmall] = ki[ksmall] / 10.0
hn[ix[ksmall]] = ki[ksmall] - knew[ksmall]
k[ix] = knew
# disp(['Iteration ',num2str(count),' Number of points left: '
# num2str(length(ix)) ]),
ix = find((abs(hn) > sqrt(eps) * abs(k)) * abs(hn) > sqrt(eps))
count += 1
if count == count_limit:
warnings.warn('W2K did not converge. The maximum error in the ' +
'last step was: %13.8f' % max(hn[ix]))
k.shape = oshape
k2 = k * sin(th)
k1 = k * cos(th)
return k1, k2
def getBeta(y):
return -(4 * (H**2) * P) / (L * (pi**3)) * np.sum(
(-1)**((n - 1) / 2) / (n**3) * (1 - np.cosh(
(n * pi * y) / H) / np.cosh((n * pi * W) / (2 * H))),
axis=0)
class SinhFunction(Function):
VJP_ALL = [
lambda g, ans, x: g * np.cosh(x)
]
# Select site to change, coordinate, and up or down
site = np.random.randint(0, size, (1, 3))[0]
coord = np.random.randint(0, 3)
e_coord = [1, 0, 0] if coord == 0 else [
0, 1, 0
] if coord == 1 else [0, 0, 1]
change = np.random.randint(0, 2)
change = -1 if change == 0 else 1
# change in energy from site change
delta_e = energy_diff(lattice, site,
np.array(e_coord) * change)
#print(delta_e)
# transition probability
prob = 1 / 2 * 1. / np.cosh(delta_e * beta / 2) * np.exp(
beta / 2)
if (prob > np.random.random()):
#print("TRANSITIONING")
#print(prob)
lattice[site[0]][site[1]][site[2]][coord] += change
acc += 1
# calculation of ground state occupancy
#gsoccupancy = sum(np.shape(~lattice.any(axis=3))) / num_atoms
gsoccupancy = np.count_nonzero(
np.sum(lattice, axis=3) == 0)
gs_occ.append(gsoccupancy / num_atoms)
# generates array of Booleans (True if it exists in ground state,
# False otherwise)
# and sums for ground state occupancy
def Pt(U0, E, L, betac, gamma_sqc):
"""return tunneling probability for square barrier"""
return 1/ (np.cosh(betac * L)**2 + gamma_sqc * np.sinh(betac * L)**2)
def sech2(x):
return np.cosh(x)**-2
def np_hyperbolic():
print(np.sinh(np.pi / 2))
print(np.cosh(np.array([np.pi / 2, np.pi / 3, np.pi / 4, np.pi / 5])))
print(np.arcsinh(1.0))
print(np.arctanh(np.array([0.1, 0.2, 0.5])))
class TanhFunction(Function):
VJP_ALL = [
lambda g, ans, x: g / np.cosh(x) ** 2
]
def test_backward_derivative_sinh(self):
# Compute the derivative of a function using a backward difference scheme
# And a backward scheme will only look below x0.
dsinh = nd.Derivative(np.sinh, method='backward')
small = np.abs(dsinh(0.0) - np.cosh(0.0)) < dsinh.error_estimate
self.assertTrue(small)
def __call__(self,phases,log10_ens=3):
e,gamma,loc = self._make_p(log10_ens)
z = TWOPI*(phases-loc)
# NB -- numpy call not as efficient as math.sinh etc.
# but this allows easy inheritance for the energy-dependence
return np.sinh(gamma)/(np.cosh(gamma)-np.cos(z))
signal.append(0)
#print 'jet#:',ijet,'etacenter:',event.JetAK8_ETA[ijet],'phicenter',event.JetAK8_PHI[ijet]
#print len(event.CaloTower_ETA)
for calo in range(len(event.CaloTower_ET)):
etadistance = int(
abs(event.CaloTower_ETA[calo] - etacenter) / .0714 + .5)
phidistance = int(
abs(event.CaloTower_PHI[calo] - phicenter) / (m.pi / 44) + .5)
# E=event.CaloTower_ETA[calo]-etacenter
#P= event.CaloTower_PHI[calo]-phicenter
#count+=1
if (etacenter >= event.CaloTower_ETA[calo]
and phicenter >= event.CaloTower_PHI[calo]) and (
etadistance <= 12 and phidistance <= 12):
Jetarray[12 - etadistance][
12 - phidistance] += event.CaloTower_ET[calo] / np.cosh(
event.CaloTower_ETA[calo])
#print event.CaloTower_ET[calo]/np.cosh(event.CaloTower_ETA[calo),
#print event.CaloTower_ET[calo]/np.cosh(event.CaloTower_ETA[calo])
elif (etacenter >= event.CaloTower_ETA[calo]
and phicenter < event.CaloTower_PHI[calo]) and (
etadistance <= 12 and phidistance <= 12):
Jetarray[12 - etadistance][
12 + phidistance] += event.CaloTower_ET[calo] / np.cosh(
event.CaloTower_ETA[calo])
#continue
elif (etacenter < event.CaloTower_ETA[calo]
and phicenter >= event.CaloTower_PHI[calo]) and (
etadistance <= 12 and phidistance <= 12):
Jetarray[12 + etadistance][
12 - phidistance] += event.CaloTower_ET[calo] / np.cosh(
event.CaloTower_ETA[calo])
def _smear_particles(particles, energy_resolutions, pt_resolutions,
eta_resolutions, phi_resolutions):
""" Applies smearing function to particles of one event """
# No smearing if any argument is None
if energy_resolutions is None or pt_resolutions is None or eta_resolutions is None or phi_resolutions is None:
return particles
smeared_particles = []
for particle in particles:
pdgid = particle.pdgid
if (pdgid not in six.iterkeys(energy_resolutions)
or pdgid not in six.iterkeys(pt_resolutions)
or pdgid not in six.iterkeys(eta_resolutions)
or pdgid not in six.iterkeys(phi_resolutions)):
continue
if None in energy_resolutions[pdgid] and None in pt_resolutions[pdgid]:
raise RuntimeError(
"Cannot derive both pT and energy from on-shell conditions!")
# Minimum energy and pT
m = particle.m
min_e = 0.0
if None in pt_resolutions[pdgid]:
min_e = m
min_pt = 0.0
# Smear four-momenta
e = None
if None not in energy_resolutions[pdgid]:
e = min_e - 1.0
while e <= min_e:
e = _smear_variable(particle.e, energy_resolutions, pdgid)
pt = None
if None not in pt_resolutions[pdgid]:
pt = min_pt - 1.0
while pt <= min_pt:
pt = _smear_variable(particle.pt, pt_resolutions, pdgid)
eta = _smear_variable(particle.eta, eta_resolutions, pdgid)
phi = _smear_variable(particle.phi(), phi_resolutions, pdgid)
while phi > 2.0 * np.pi:
phi -= 2.0 * np.pi
while phi < 0.0:
phi += 2.0 * np.pi
# Construct particle
smeared_particle = MadMinerParticle()
if None in energy_resolutions[pdgid]:
# Calculate E from on-shell conditions
smeared_particle.setptetaphim(pt, eta, phi, particle.m)
elif None in pt_resolutions[pdgid]:
# Calculate pT from on-shell conditions
if e > m:
pt = (e**2 - m**2)**0.5 / np.cosh(eta)
else:
pt = 0.0
smeared_particle.setptetaphie(pt, eta, phi, e)
else:
# Everything smeared manually
smeared_particle.setptetaphie(pt, eta, phi, e)
# PDG id (also sets charge)
smeared_particle.set_pdgid(pdgid)
smeared_particles.append(smeared_particle)
return smeared_particles