def _R2deriv(self, R, z, phi=0., t=0.):
"""
NAME:
_Rderiv
PURPOSE:
evaluate the second radial derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the second radial derivative
HISTORY:
2018-08-04 - Written - Bovy (UofT)
"""
Raz2 = (R + self.a)**2 + z**2
Raz = numpy.sqrt(Raz2)
m = 4. * R * self.a / Raz2
R2ma2mz2o4aR1m = (R**2 - self.a2 - z**2) / 4. / self.a / R / (1. - m)
return (2*R**2+self.a2+3*R*self.a+z**2)/R/Raz2*self._Rforce(R,z)\
+2.*self.a/R/Raz*(m*(R**2+self.a2+z**2)/4./(1.-m)/self.a/R**2\
*special.ellipe(m)\
+(R2ma2mz2o4aR1m/(1.-m)*special.ellipe(m)
+0.5*R2ma2mz2o4aR1m*(special.ellipe(m)-special.ellipk(m))
+0.5*(special.ellipe(m)/(1.-m)-special.ellipk(m))/m)\
*4*self.a*(self.a2+z**2-R**2)/Raz2**2)
def action_from_oscillation_amplitude(RFStation,
dtmax,
timestep=0,
Np_histogram=None):
'''
Returns the relative action for given oscillation amplitude in time,
assuming single-harmonic RF system and no intensity effects.
Action is normalised to the value at the separatrix, given in units of 1.
Optional: RF parameters at a given timestep (default = 0) are used.
Optional: Number of points for histogram output
'''
omega_rf = RFStation.omega_RF[0, timestep]
xx = x2(omega_rf * dtmax)
action = np.zeros(len(xx))
indices = np.where(xx != 1.)[0]
indices0 = np.where(xx == 1.)[0]
action[indices] = (ellipe(xx[indices]) -
(1. - xx[indices]) * ellipk(xx[indices]))
if indices0:
action[indices0] = np.float(ellipe(xx[indices0]))
if Np_histogram != None:
histogram, bins = np.histogram(action, Np_histogram, (0, 1))
histogram = np.double(histogram) / np.sum(histogram[:])
bin_centres = 0.5 * (bins[0:-1] + bins[1:])
return action, bin_centres, histogram
else:
return action
def calc_PrimaryRegion(self, X, Z):
"""Predicts magnitude and direction of primary field in region"""
# CALCULATES INDUCING FIELD WITHIN REGION AND RETURNS AT LOCATIONS
# Initiate Variables from object
I = self.I
a1 = self.a1
eps = 1e-6
mu0 = 4 * np.pi * 1e-7 # 1e9*mu0
s = np.abs(X) # Define Radial Distance
k = 4 * a1 * s / (Z**2 + (a1 + s)**2)
Bpx = mu0 * np.sign(X) * (Z * I / (2 * np.pi * s + eps)) * (
1 / np.sqrt(Z**2 + (a1 + s)**2)) * (-sp.ellipk(k) +
((a1**2 + Z**2 + s**2) /
(Z**2 +
(s - a1)**2)) * sp.ellipe(k))
Bpz = mu0 * (I / (2 * np.pi)) * (1 / np.sqrt(Z**2 + (a1 + s)**2)) * (
sp.ellipk(k) + ((a1**2 - Z**2 - s**2) /
(Z**2 + (s - a1)**2)) * sp.ellipe(k))
Bpx[(X > -1.025 * a1) & (X < -0.975 * a1) & (Z > -0.025 * a1) &
(Z < 0.025 * a1)] = 0.
Bpx[(X < 1.025 * a1) & (X > 0.975 * a1) & (Z > -0.025 * a1) &
(Z < 0.025 * a1)] = 0.
Bpz[(X > -1.025 * a1) & (X < -0.975 * a1) & (Z > -0.025 * a1) &
(Z < 0.025 * a1)] = 0.
Bpz[(X < 1.025 * a1) & (X > 0.975 * a1) & (Z > -0.025 * a1) &
(Z < 0.025 * a1)] = 0.
Babs = np.sqrt(Bpx**2 + Bpz**2)
return Bpx, Bpz, Babs
def _R2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
_Rderiv
PURPOSE:
evaluate the second radial derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the second radial derivative
HISTORY:
2018-08-04 - Written - Bovy (UofT)
"""
Raz2= (R+self.a)**2+z**2
Raz= nu.sqrt(Raz2)
m= 4.*R*self.a/Raz2
R2ma2mz2o4aR1m= (R**2-self.a2-z**2)/4./self.a/R/(1.-m)
return (2*R**2+self.a2+3*R*self.a+z**2)/R/Raz2*self._Rforce(R,z)\
+2.*self.a/R/Raz*(m*(R**2+self.a2+z**2)/4./(1.-m)/self.a/R**2\
*special.ellipe(m)\
+(R2ma2mz2o4aR1m/(1.-m)*special.ellipe(m)
+0.5*R2ma2mz2o4aR1m*(special.ellipe(m)-special.ellipk(m))
+0.5*(special.ellipe(m)/(1.-m)-special.ellipk(m))/m)\
*4*self.a*(self.a2+z**2-R**2)/Raz2**2)
def Int3(q, r, R):
'''
Integral I3.
'''
m = 4 * R * r / (q**2 + (r + R)**2)
K0 = ellipk(m) # Complete elliptic integral of the first kind
K1 = ellipk(1 - m)
E0 = ellipe(m) # Complete elliptic integral of the second kind
E1 = ellipe(1 - m)
beta = np.arcsin(q / np.sqrt(q**2 + (R - r)**2))
K2 = ellipkinc(beta,
1 - m) # Incomplete elliptic integral of the first kind
E2 = ellipeinc(beta,
1 - m) # Incomplete elliptic integral of the second kind
Z = E2 - E1 * K2 / K1 # Jacobi zeta function
lamb = K2 / K1 + 2 * K0 * Z / np.pi # Heuman’s lambda function
I3 = -q * np.sqrt(m) * K0 / (2 * np.pi * R * np.sqrt(r * R)) + (
np.heaviside(r - R, 0.5) - np.heaviside(R - r, 0.5)) * lamb / (
2 * R) + np.heaviside(R - r, 0.5) / R
return I3
def action_from_oscillation_amplitude(RFStation, dtmax, timestep = 0,
Np_histogram = None):
'''
Returns the relative action for given oscillation amplitude in time,
assuming single-harmonic RF system and no intensity effects.
Action is normalised to the value at the separatrix, given in units of 1.
Optional: RF parameters at a given timestep (default = 0) are used.
Optional: Number of points for histogram output
'''
omega_rf = RFStation.omega_RF[0,timestep]
xx = x2(omega_rf*dtmax)
action = np.zeros(len(xx))
indices = np.where(xx != 1.)[0]
indices0 = np.where(xx == 1.)[0]
action[indices] = (ellipe(xx[indices]) -
(1. - xx[indices])*ellipk(xx[indices]))
if indices0:
action[indices0] = np.float(ellipe(xx[indices0]))
if Np_histogram != None:
histogram, bins = np.histogram(action, Np_histogram, (0,1))
histogram = np.double(histogram)/np.sum(histogram[:])
bin_centres = 0.5*(bins[0:-1] + bins[1:])
return action, bin_centres, histogram
else:
return action
def calc_G(lamda, mu, alpha, r):
'''
Calculates the G transform for a disk of radius alpha at a distance r, on top of a substrate with Lame parameter lambda and shear modulus mu.
lamda: Lame parameter of substrate
mu: shear modulus of substrate
alpha: disk radius, in metres
r: array of distances from centre of disk at which to calculate solution. In metres. eg r = np.linspace(0,50*10**3,num=1000) to go to 50km distance.
'''
sigma = lamda + 2 * mu
nabla = lamda + mu
defm = np.zeros_like(r)
r_disk = r[r <= alpha]
r_postdisk = r[r >= alpha]
defm[r <= alpha] = g * (sigma /
(np.pi**2 * mu * nabla * alpha) * special.ellipe(
(r_disk / alpha)**2))
defm[r >= alpha] = g * (sigma * r_postdisk /
(np.pi**2 * mu * nabla * alpha**2)) * (
special.ellipe((alpha / r_postdisk)**2) -
(1 - (alpha / r_postdisk)**2) * special.ellipk(
(alpha / r_postdisk)**2))
return defm
def findCap(height, eff):
coeffInSideBracketsa = (math.pi * xa) / (2 * height)
coeffInSideBracketsb = (math.pi * xb) / (2 * height)
coeffInSideBracketsc = (math.pi * xc) / (2 * height)
coeffa = math.sinh(coeffInSideBracketsa)
coeffasquared = math.pow(coeffa, 2)
coeffb = math.sinh(coeffInSideBracketsb)
coeffbsquared = math.pow(coeffb, 2)
coeffc = math.sinh(coeffInSideBracketsc)
coeffcsquared = math.pow(coeffc, 2)
kp1 = coeffc / coeffb
kInsideSqurt = (coeffbsquared - coeffasquared) / (coeffcsquared -
coeffasquared)
kp2 = math.sqrt(kInsideSqurt)
k = kp1 * kp2
ksquared = math.pow(k, 2)
kder = math.sqrt(1 - ksquared)
K = ellipe(k)
Kder = ellipe(kder)
Kcoeff = Kder / K
C = 2 * relativePermittivityOfFreeSpace * eff * Kcoeff
return C
def _A_impl(self, A):
points = self.get_points_cart_ref()
points = np.array(
np.dot(self.rotMatrix,
np.array(np.subtract(points, self.center)).T).T)
rho = np.sqrt(np.square(points[:, 0]) + np.square(points[:, 1]))
r = np.sqrt(
np.square(points[:, 0]) + np.square(points[:, 1]) +
np.square(points[:, 2]))
alpha = np.sqrt(self.r0**2 + np.square(r) - 2 * self.r0 * rho)
beta = np.sqrt(self.r0**2 + np.square(r) + 2 * self.r0 * rho)
k = np.sqrt(1 - np.divide(np.square(alpha), np.square(beta)))
ellipek2 = ellipe(k**2)
ellipkk2 = ellipk(k**2)
A[:] = -self.Inorm / 2 * np.dot(
self.rotMatrixInv,
np.array(
(2 * self.r0 +
np.sqrt(points[:, 0]**2 + points[:, 1]**2) * ellipek2 +
(self.r0**2 + points[:, 0]**2 + points[:, 1]**2 +
points[:, 2]**2) * (ellipe(k**2) - ellipkk2)) /
((points[:, 0]**2 + points[:, 1]**2) *
np.sqrt(self.r0**2 + points[:, 0]**2 + points[:, 1]**2 + 2 *
self.r0 * np.sqrt(points[:, 0]**2 + points[:, 1]**2) +
points[:, 2]**2)) *
np.array([-points[:, 1], points[:, 0], 0])).T)
def calc_PrimaryLoop(self):
"""Predicts magnitude and direction of primary field in loop center"""
# CALCULATES INDUCING FIELD AT RX LOOP CENTER
# Initiate Variables
I = self.I
a1 = self.a1
x = self.x
z = self.z
eps = 1e-7
mu0 = 4 * np.pi * 1e-7 # 1e9*mu0
s = np.abs(x) # Define Radial Distance
k = 4 * a1 * s / (z**2 + (a1 + s)**2)
Bpx = mu0 * np.sign(x) * (z * I / (2 * np.pi * s + eps)) * (
1 / np.sqrt(z**2 + (a1 + s)**2)) * (-sp.ellipk(k) +
((a1**2 + z**2 + s**2) /
(z**2 +
(s - a1)**2)) * sp.ellipe(k))
Bpz = mu0 * (I / (2 * np.pi)) * (1 / np.sqrt(z**2 + (a1 + s)**2)) * (
sp.ellipk(k) + ((a1**2 - z**2 - s**2) /
(z**2 + (s - a1)**2)) * sp.ellipe(k))
self.Bpx = Bpx
self.Bpz = Bpz
def f2D(k, k_fermi):
"""Return the value of the function f2D from Guiliani and Vignale, pg 81."""
y = np.linalg.norm(k) / k_fermi
if y <= 1.0:
return special.ellipe(y)
else:
return y * (special.ellipe(1.0 / y) -
(1.0 - (1.0 / y**2)) * special.ellipk(1.0 / y))
def cal_cp(Dw, alpha0, Dpw, ri, re, Ep, ve, tol=0.001):
"""
calculation of the spring constant cp, acc. to ISO 16218 Function 11.
Args:
tol: tolerance for convergence
Dw: diameter of the ball
alpha0: initial contact angle
Dpw: pitch diameter of the bearing
ri: cross-sectional raceway groove radius, inner
re: cross-sectional raceway groove radius, outer
Ep: modulus of elasticity
ve: poisson's ratio
Returns:
cp: float
"""
gamma = Dw * cos(alpha0) / Dpw
# 内外圈曲率和
rho_i = 2 / Dw * (2 + gamma / (1 - gamma) - Dw / 2 / ri)
rho_e = 2 / Dw * (2 - gamma / (1 + gamma) - Dw / 2 / re)
# 内外圈曲率差
Fip = (gamma / (1 - gamma) + Dw / 2 / ri) / (2 + gamma /
(1 - gamma) - Dw / 2 / ri)
Fep = (-gamma / (1 + gamma) + Dw / 2 / re) / (2 - gamma /
(1 + gamma) - Dw / 2 / ri)
for k in arange(0, 1, 0.001): # 内圈迭代求解
if k == 0:
chi = float("inf")
else:
chi = 1 / k
M = 1 - 1 / chi**2
# 第一类和第二类椭圆积分
Ki = ellipk(M)
Ei = ellipe(M)
Fp = 1 - 2 / (chi**2 - 1) * (Ki / Ei - 1)
if abs((Fp - Fip) / Fp) < tol:
chi_i = chi
break
else:
pass
for k in arange(0, 1, 0.001): # 外圈迭代求解
if k == 0:
chi = float("inf")
else:
chi = 1 / k
M = 1 - 1 / chi**2
# 第一类和第二类椭圆积分
Ke = ellipk(M)
Ee = ellipe(M)
Fp = 1 - 2 / (chi**2 - 1) * (Ke / Ee - 1)
if abs((Fp - Fep) / Fp) < tol:
chi_e = chi
break
else:
pass
return (1.48 * Ep / (1 - ve**2) *
((Ki * (rho_i / chi_i**2 / Ei)**(1 / 3) + Ke *
(rho_e / chi_e**2 / Ee)**(1 / 3)))**(-1.5))
def surface_displacement(x, y, transform=0):
"""
Into surface displacement at the surface for an elliptical contact
Parameters
----------
x,y : array-like
x and y coordinates of the points of interest
transform : int {0,1,2}, optional (0)
a flag which defines which axes the result is displayed on. If set
to 0 the result is displayed on the 'contact axes' which are
aligned with the principal radii of the contact ellipse. If set to
1 or 2 the result is aligned with the axes of the first or second
body respectively.
Returns
-------
displacement : array
The into surface displacement at each of the points of interest
Notes
-----
The pressure distribution is given by:
p(x,y)=p0*(1-(x/a)**2-(y/b)**2)**0.5
References
----------
[1] Johnson, K. (1985). Contact Mechanics. Cambridge: Cambridge
University Press. doi:10.1017/CBO9781139171731
"""
nonlocal l_johnson, m_johnson, n_johnson
if l_johnson is None:
if b > a:
raise ValueError("Change in a>b or b>a between sources, "
"sort out")
e = (1 - b**2 / a**2)**0.5
l_johnson = np.pi * p0 * b * special.ellipk(e)
m_johnson = np.pi * p0 * b / e**2 / a**2 * (special.ellipk(e) -
special.ellipe(e))
n_johnson = np.pi * p0 * b / a**2 / e**2 * (
(a**2 / b**2) * special.ellipe(e) - special.ellipk(e))
if transform:
x, y = _transform_axes(x, y, [alpha, beta][transform - 1])
out_of_bounds = np.clip((1 - (x / a)**2 - (y / b)**2), 0,
float('inf')) == 0
displacement = np.array(
(1 - v**2) / modulus / np.pi *
(l_johnson - m_johnson * x**2 - n_johnson * y**2))
displacement[out_of_bounds] = float('Nan')
return displacement
def EllipticE():
"""Return EllipticE."""
if (b == 0):
return 0
elif (ksq == 1):
return 1
elif (ksq < 1):
return ellipe(ksq)
else:
return ellipe(1. / ksq)
def a_hauptmanmiloh(A):
"""Calculate the wing lift slope using Hauptman and Miloh's approximation
This function calculates the wing lift slope of a finite elliptic wing
using the equation
a = \left\{\begin{matrix} \left[ \frac{E^2(h)}{\pi A + (4k^3/h)
\ln(1/k + h/k)} + \frac{1}{\pi A} \right]^{-1}, &
k=\frac{\pi A}{4} \le 1 \\ \left[ \frac{E^2(h)}{4 (k +
\sin^{-1}(h)/h)} + \frac{1}{\pi A} \right]^{-1}, &
k=\frac{4}{\pi A} \le 1 \end{matrix}\right.
which corresponds to the approximation given by Hauptman and Miloh in 1986.
Note that a0 does not appear in the equations because they were derived
explicitly for a flate plate with a0 = 2*pi.
Inputs:
A = Aspect ratio of wing (b^2 / Sw)
"""
if A < (4.0 / np.pi):
# Slender wing
# Calculate the eccentricity and parameter for the ellipse
k = (np.pi * A) / 4.0
h = np.sqrt(1.0 - k**2)
# Calculate the perimeter of the ellipse using the complete elliptic
# integral of the second kind
E = ellipe(h**2)
# Calculate the resistance value r1
r1 = (np.pi * A + (4.0 * k**3 / h * np.log((1.0 + h) / k))) / E**2
elif A > (4.0 / np.pi):
# High-aspect-ratio wing
# Calculate the eccentricity and parameter for the ellipse
k = 4.0 / (np.pi * A)
h = np.sqrt(1.0 - k**2)
# Calculate the perimeter of the ellipse using the complete elliptic
# integral of the second kind
E = ellipe(h**2)
# Calculate the resistance value r1
r1 = 4 * (k + np.arcsin(h) / h) / E**2
else:
# Circular wing
r1 = 32.0 / (2.0 + np.pi**2)
# Calculate the resistance value r2
r2 = np.pi * A
# Calculate the wing lift slope using the parallel resistors method
return resistance_parallel(r1, r2)
def loop_current(a, I, rho, z):
alphasq = a * a + rho * rho + z * z - 2 * a * rho
betasq = a * a + rho * rho + z * z + 2 * a * rho
beta = sqrt(betasq)
ksq = 1. - alphasq / betasq
C = mu0 * I / pi
Brho = (C * z / (2 * alphasq * rho * beta)) * (
(a * a + rho * rho + z * z) * ellipe(ksq) - alphasq * ellipk(ksq))
Bz = (C / (2 * alphasq * beta)) * (
(a * a - rho * rho - z * z) * ellipe(ksq) + alphasq * ellipk(ksq))
return Brho, Bz
def findC0(xa, xb, xc):
xasquared = xa**2
xbsquared = math.pow(xb, 2)
xcsquared = math.pow(xc, 2)
kp1 = xc / xb
kInsideSqurt = (xbsquared - xasquared) / (xcsquared - xasquared)
kp2 = math.sqrt(kInsideSqurt)
k = kp1 * kp2
ksquared = math.pow(k, 2)
kder = math.sqrt(1 - ksquared)
K = ellipe(k)
Kder = ellipe(kder)
C0 = (4 * relativePermittivityOfFreeSpace * Kder) / K
return C0
def calc_PrimaryLoop(self):
"""Predicts magnitude and direction of primary field in loop center"""
# CALCULATES INDUCING FIELD AT RX LOOP CENTER
# Initiate Variables
I = self.I
a1 = self.a1
a2 = self.a2
x = self.x
z = self.z
azm = self.azm
eps = 1e-7
mu0 = 4 * np.pi * 1e-7 # 1e9*mu0
s = np.abs(x) # Define Radial Distance
k = 4 * a1 * s / (z ** 2 + (a1 + s) ** 2)
Bpx = (
mu0
* np.sign(x)
* (z * I / (2 * np.pi * s + eps))
* (1 / np.sqrt(z ** 2 + (a1 + s) ** 2))
* (
-sp.ellipk(k)
+ ((a1 ** 2 + z ** 2 + s ** 2) / (z ** 2 + (s - a1) ** 2))
* sp.ellipe(k)
)
)
Bpz = (
mu0
* (I / (2 * np.pi))
* (1 / np.sqrt(z ** 2 + (a1 + s) ** 2))
* (
sp.ellipk(k)
+ ((a1 ** 2 - z ** 2 - s ** 2) / (z ** 2 + (s - a1) ** 2))
* sp.ellipe(k)
)
)
Bpabs = np.sqrt(Bpx ** 2 + Bpz ** 2)
Bpn = np.sin(np.deg2rad(azm)) * Bpx + np.cos(np.deg2rad(azm)) * Bpz
Area = np.pi * a2 ** 2
self.Bpx = Bpx
self.Bpz = Bpz
self.Bpabs = Bpabs
self.Bpn = Bpn
self.Area = Area
def test_comp_ellint_2():
if NumCpp.NO_USE_BOOST and not NumCpp.STL_SPECIAL_FUNCTIONS:
return
a = np.random.rand(1).item()
assert (roundScaler(NumCpp.comp_ellint_2_Scaler(a), NUM_DECIMALS_ROUND) ==
roundScaler(sp.ellipe(a**2).item(), NUM_DECIMALS_ROUND))
shapeInput = np.random.randint(20, 100, [2, ])
shape = NumCpp.Shape(shapeInput[0].item(), shapeInput[1].item())
aArray = NumCpp.NdArray(shape)
a = np.random.rand(shape.rows, shape.cols)
aArray.setArray(a)
assert np.array_equal(roundArray(NumCpp.comp_ellint_2_Array(aArray), NUM_DECIMALS_ROUND),
roundArray(sp.ellipe(np.square(a)), NUM_DECIMALS_ROUND))
def local(self, R):
# RCoil=[np.array([1.0,0.0])]
r = np.sqrt(R[0]**2 + R[1]**2)
z1 = R[2]
theta = np.arctan(R[1] / R[0])
Br = 0.0
Bz = 0.0
BrR = 0.0
for n in range(len(self.RCoil)):
r0 = self.RCoil[n][0]
z0 = self.RCoil[n][1]
z = z1 - z0
k = np.sqrt(4 * r * r0 / ((r + r0)**2 + z**2))
IE = sp.ellipe(k)
IK = sp.ellipk(k)
Br = Br + (1.0 / r) * (z / np.sqrt((r + r0)**2 + z**2)) * \
(-IK + IE * (r0**2 + r**2 + z**2) / ((r0 - r)**2 + z**2))
Bz = Bz + (1.0 / np.sqrt((r + r0)**2 + z**2)) * \
(IK + IE * (r0**2 - r**2 - z**2) / ((r0 - r)**2 + z**2))
if ((r - r0)**2 + z**2 < 0.1**2):
Br = 0
Bz = 0
break
BV = (mu * self.I0) / (2.0 * r0) / self.nCoil * \
np.array([Br * np.cos(theta), Br * np.sin(theta), Bz])
return BV
def ellippi(n, m):
"""
Complete elliptic integral of third kind (simplified version).
.. FIXME: Incorrect, because of non-standard definitions of elliptic integral in the reference
Reference
---------
F. LAMARCHE and C. LEROY, Evaluation of the volume of a sphere with a cylinder by elliptic integrals,
Computer Phys. Comm. 59 (1990) pg. 365
"""
a2 = n
k2 = m
k2_a2 = k2 + a2
phi = np.arcsin(a2 / k2_a2)
Kc = ellipk(k2)
Ec = ellipe(k2)
Ki = ellipkinc(phi, (1. - k2) ** 1)
Ei = ellipeinc(phi, (1. - k2) ** 1)
c1 = k2 / k2_a2
c2 = np.sqrt(a2 / (1 + a2) / k2_a2)
return c1 * Kc + c2 * ((Kc - Ec) * Ki + Kc * Ei)
def period(self):
'''Analytically calculate the period of EKM oscillations.'''
# First calculate the limits.
xcrit = brentq(lambda x: ellipk(x) - 2 * ellipe(x), 0, 1)
phicrit = 3 * (1 - xcrit) / (3 + 2 * xcrit)
if self.phiq < phicrit:
CKLmin = brentq(lambda CKL: self.chi - self.epsoct - F(CKL), self.tol, self.phiq)
else:
# Check if flips occur for Omega = Pi or 0
if (np.sign(self.chi - self.epsoct - F(self.tol)) !=
np.sign(self.chi - self.epsoct - F(self.phiq))):
CKLmin = brentq(lambda CKL: self.chi - self.epsoct - F(CKL), self.tol, self.phiq)
else:
CKLmin = brentq(lambda CKL: self.chi + self.epsoct - F(CKL), self.tol, self.phiq)
if self.doesflip():
CKLmax = self.phiq
else:
CKLmax = brentq(lambda CKL: self.chi + self.epsoct - F(CKL), 0, 1)
prefactor = 256 * np.sqrt(10) / (15 * np.pi) / self.epsoct
P = quad(lambda CKL: (prefactor * ellipk((3 - 3*CKL)/(3 + 2*CKL)) /
(4 - 11*CKL) / np.sqrt(6 + 4*CKL) / np.sqrt(1 - 1/self.epsoct**2 *
(F(CKL) - self.chi)**2) / np.sqrt(2* np.fabs(self.phiq - CKL))),
CKLmin, CKLmax, epsabs=1e-12, epsrel=1e-12, limit=100)
return P[0]
def cnoidalwaves(x, t, dx, a0, a1, g, k):
n = len(x)
u = zeros(n)
h = zeros(n)
bed = zeros(n)
m = k * k
h0 = a0 + a1 * (float(ellipe(m)) / ellipk(m))
c = sqrt((g * a0 * (a0 + a1) * (a0 + (1 - k * k) * a1))) / float(h0)
Kc = sqrt(float(3 * a1) / (4 * a0 * (a0 + a1) * (a0 + (1 - k * k) * a1)))
for i in range(n):
h[i] = a0 + a1 * dnsq(Kc * (x[i] - c * t), m)
u[i] = c * (1 - float(h0) / h[i])
h0i = a0 + a1 * dnsq(Kc * (x[0] - dx - c * t), m)
u0i = c * (1 - float(h0) / h0i)
h1i = a0 + a1 * dnsq(Kc * (x[-1] + dx - c * t), m)
u1i = c * (1 - float(h0) / h1i)
G = getGfromupy(h, u, bed, u0i, u1i, h0i, h1i, bed[0], bed[-1], dx)
return h, u, G, bed
def ring_force():
s = rp - rq
a = rp + rq
four = -4 * (rp * rq) / s**2
val = (1 / (2 * pi)) * q_q * q_p * (2 / (rp * np.abs(s) * a)) * (s * sp.ellipe(four) + a * \
sp.ellipk(four))
return val
def ellippi(n, m):
"""
Complete elliptic integral of third kind (simplified version).
.. FIXME: Incorrect, because of non-standard definitions of elliptic integral in the reference
Reference
---------
F. LAMARCHE and C. LEROY, Evaluation of the volume of a sphere with a cylinder by elliptic integrals,
Computer Phys. Comm. 59 (1990) pg. 365
"""
a2 = n
k2 = m
k2_a2 = k2 + a2
phi = np.arcsin(a2 / k2_a2)
Kc = ellipk(k2)
Ec = ellipe(k2)
Ki = ellipkinc(phi, (1. - k2)**1)
Ei = ellipeinc(phi, (1. - k2)**1)
c1 = k2 / k2_a2
c2 = np.sqrt(a2 / (1 + a2) / k2_a2)
return c1 * Kc + c2 * ((Kc - Ec) * Ki + Kc * Ei)
def _B_impl(self, B):
points = self.get_points_cart_ref()
points = np.array(
np.dot(self.rotMatrix,
np.array(np.subtract(points, self.center)).T).T)
rho = np.sqrt(np.square(points[:, 0]) + np.square(points[:, 1]))
r = np.sqrt(
np.square(points[:, 0]) + np.square(points[:, 1]) +
np.square(points[:, 2]))
alpha = np.sqrt(self.r0**2 + np.square(r) - 2 * self.r0 * rho)
beta = np.sqrt(self.r0**2 + np.square(r) + 2 * self.r0 * rho)
k = np.sqrt(1 - np.divide(np.square(alpha), np.square(beta)))
ellipek2 = ellipe(k**2)
ellipkk2 = ellipk(k**2)
gamma = np.square(points[:, 0]) - np.square(points[:, 1])
B[:] = np.dot(
self.rotMatrixInv,
np.array([
self.Inorm * points[:, 0] * points[:, 2] /
(2 * alpha**2 * beta * rho**2) *
((self.r0**2 + r**2) * ellipek2 - alpha**2 * ellipkk2),
self.Inorm * points[:, 1] * points[:, 2] /
(2 * alpha**2 * beta * rho**2) *
((self.r0**2 + r**2) * ellipek2 - alpha**2 * ellipkk2),
self.Inorm / (2 * alpha**2 * beta) *
((self.r0**2 - r**2) * ellipek2 + alpha**2 * ellipkk2)
])).T
def strains_lat(eps_22, eps_33, r0):
# computes lateral strains for a 0 deg triax test (vertical bedding planes),
# based on strains in 22- and 33-direction
# compute initial circumference
u0 = 2. * np.pi * r0
# compute main radii of ellipsis, based on the strains
r_22 = r0 * (1. + eps_22)
r_33 = r0 * (1. + eps_33)
# check which radius is larger and assign it to a and the smaller one to b
if r_22 >= r_33:
a = r_22
b = r_33
else:
a = r_33
b = r_22
# compute elliptical circumference of deformed section
aux_e = 1. - b**2 / a**2
u_ellipsis = 4. * a * special.ellipe(aux_e)
# compute lateral strains, based on the elliptical circumference and return it
return u_ellipsis / u0 - 1.
def sinCurve(plotOrNot):
nPoints = 120;
#The length of the sine curve y = Asin(kx) x in [0,pi/(2k)]
#is E(-A^2k^2)/k E is the Complete elliptic integral of the second kind
#can be evaluated as scipy.special.ellipe(m)
A = 0.0971;
k = 4;
xr = 0.15
l = scispe.ellipe(-A**2*(k*np.pi/xr)**2)/(k*np.pi/xr)*(4*k)
print('sine curve length is', l)
xArray = np.linspace(-xr,xr,num=nPoints)
yArray = A*(np.sin(k*np.pi/xr*(xArray + xr) - np.pi/2.0) + 1.0)
dl = 0.0
max_ddl,min_ddl = 0.0,1.0
for i in range(len(xArray) - 1):
ddl = np.sqrt((xArray[i+1] - xArray[i])**2 + (yArray[i+1] - yArray[i])**2)
if ddl > max_ddl:
max_ddl = ddl
if ddl < min_ddl:
min_ddl = ddl
dl+= ddl
print('discrete sine curve length is', dl, 'max segment length is ', max_ddl, 'min segment length is ', min_ddl)
if (plotOrNot):
plt.plot(xArray, yArray, '-*')
plt.ylim([-0.5, 1.5])
plt.xlim([-1, 1])
plt.show()
return nPoints, xArray, yArray
def cnoidalwaves(x,t,dx,a0,a1,g,k):
n = len(x)
u = zeros(n)
h = zeros(n)
bed = zeros(n)
m = k*k
h0 = a0 + a1*(float(ellipe(m)) / ellipk(m))
c = sqrt((g*a0*(a0 + a1)*(a0 + (1 - k*k)*a1))) / float(h0)
Kc = sqrt(float(3*a1) / (4*a0*(a0 + a1)*(a0 + (1-k*k)*a1)))
for i in range(n):
h[i] = a0 + a1*dnsq(Kc*(x[i] - c*t),m)
u[i] = c *(1 - float(h0)/h[i])
h0i = a0 + a1*dnsq(Kc*(x[0] - dx - c*t),m)
u0i = c *(1 - float(h0)/h0i)
h1i = a0 + a1*dnsq(Kc*(x[-1] + dx - c*t),m)
u1i = c *(1 - float(h0)/h1i)
G = getGfromupy(h,u,bed,u0i,u1i,h0i,h1i,bed[0],bed[-1],dx)
return h,u,G,bed
def calculateGreen(self, x, xp):
if np.allclose(x, xp, rtol=1.0e-13) or math.fabs(xp[0]) < 1.0e-14:
raise FB.SingularPoint
r = x[0]
rp = xp[0]
z = x[1]
zp = xp[1]
mt = (z - zp)**2 + (r + rp)**2
m = 4 * r * rp / mt
self.G = 0.0
self.gradG[0] = 0.0
self.gradG[1] = 0.0
mt = math.sqrt(mt) * np.pi
kint = scs.ellipk(m)
eint = scs.ellipe(m)
if np.isnan(kint) or np.isnan(eint):
raise Exception('False elliptic integral m = ' + str(m))
self.G = rp * ((2.0 - m) * kint - 2.0 * eint) / (m * mt)
self.gradG[0] = r * (2.0 * m - 4.0) * kint
self.gradG[0] += r * (m * m - 8.0 * m + 8.0) / (2.0 * (1.0 - m)) * eint
self.gradG[0] += rp * m * kint
self.gradG[0] -= rp * m * (2.0 - m) / (2.0 * (1.0 - m)) * eint
self.gradG[0] /= (math.sqrt(m) * (math.sqrt(r * rp)**3))
self.gradG[0] *= rp / (4.0 * np.pi)
self.gradG[1] = (2.0 - m) / (1.0 - m) * eint - 2.0 * kint
self.gradG[1] *= math.sqrt(m) * (z - zp) / (2.0 * math.sqrt(r * rp)**3)
self.gradG[1] *= rp / (4.0 * np.pi)
def Bvector(self,r):
'''
calculate B vector (T/amp) at a point r.
convert r to coil frame. then get B then convert b
back to lab frame.
'''
x,y,z = self.posToCoil(r)
R = self.R
rho = norm([x,y])
d = np.sqrt( (R+rho)**2 + z**2 )
if d == 0: # No Coil
return np.asarray([0,0,0])
d2 = ( (R-rho)**2 + z**2 )
if d2 == 0: # on coil
return np.asarray([0,0,0])
k2 = (4*R*rho)/d**2
K = ellipk(k2)
E = ellipe(k2)
Bc = (1/d)*(K + E*(R**2 - rho**2 - z**2)/d2)
if rho == 0:
Br=Ba=Bb = 0
else:
Br = (1/rho)*(z/d)*(-K + E*(R**2 + rho**2 + z**2)/d2)
Ba = Br*x/rho
Bb = Br*y/rho
B = np.asarray([Ba,Bb,Bc])*self.unit
return self.toLab(B)
def evaluate_field_at_point(self, x, y, z):
rad = np.sqrt(x**2 + y**2)
# If on-axis
if rad / self.radius < 1e-10:
return 0.0, 0.0, self.__on_axis_field(z)
# z relative to position of coil
z_rel = z - self.z
b_central = self.__central_field()
rad_norm = rad / self.radius
z_norm = z_rel / self.radius
alpha = (1.0 + rad_norm)**2 + z_norm**2
root_alpha_pi = np.sqrt(alpha) * np.pi
beta = 4 * rad_norm / alpha
int_e = ellipe(beta)
int_k = ellipk(beta)
gamma = alpha - 4 * rad_norm
b_r = b_central * (int_e * ((1.0 + rad_norm**2 + z_norm**2) / gamma) -
int_k) / root_alpha_pi * (z_rel / rad)
b_z = b_central * (int_e * (
(1.0 - rad_norm**2 - z_norm**2) / gamma) + int_k) / root_alpha_pi
return b_r * x / rad, b_r * y / rad, b_z
def loopbrz(Ra, I0, Nturns, R, Z):
# Input
# Ra [m] Loop radius
# I0 [A] Loop current
# Nturns Loop number of turns (windings)
# R [m] Radial coordinate of the point
# Z [m] Axial coordinate of the point
# Output
# Br, Bz [T] Radial and Axial components of B-field at (R,Z)
#
# (Note that singularities are not handled here)
mu0 = 4.0e-7 * np.pi
B0 = mu0 / 2.0 / Ra * I0 * Nturns
alfa = np.absolute(R) / Ra
beta = Z / Ra
gamma = (Z + 1.0e-10) / (R + 1.0e-10)
Q = (1 + alfa)**2 + beta**2
ksq = 4.0 * alfa / Q
asq = alfa * alfa
bsq = beta * beta
Qsp = 1.0 / np.pi / np.sqrt(Q)
K = special.ellipk(ksq)
E = special.ellipe(ksq)
Br = gamma * B0 * Qsp * (E * (1 + asq + bsq) / (Q - 4.0 * alfa) - K)
Bz = B0 * Qsp * (E * (1 - asq - bsq) / (Q - 4.0 * alfa) + K)
return Br, Bz
def siglecoil_fB(I0,N_coil,r0,x,y,z):
c0=4e-7; # mu0/pi=4e-7
I0=I0*N_coil;
# current, unit: A
C=c0*I0;
# Set field for single coil
rho=(x**2+y**2)**0.5
r=(x**2+y**2+z**2)**0.5
alpha=(r0**2 + r**2 - 2*r0*rho)**0.5
beta=(r0**2 + r**2 + 2*r0*rho)**0.5
gamma=x**2-y**2
k2 = 1 - alpha**2/beta**2
(K,E)=(sp.ellipk(k2),sp.ellipe(k2))
t1=r0**4*(-gamma*(3*z**2+r0**2) + rho**2*(8*x**2-y**2))
t2=r0**2*(rho**4*(5*x**2+y**2) - 2*rho**2*z**2*(2*x**2+y**2)
+3*z**4*gamma)
t3=r**4*(2*x**4+gamma*(y**2+z**2))
t4=r0**2*( gamma*(r0**2+2*z**2) - rho**2*(3*x**2-2*y**2) )
t5=r**2*( 2*x**4 + gamma*(y**2 + z**2) )
Bxx=C*z/(2*alpha**4*beta**3*rho**4)* ( (t1-t2-t3)* E + (t4+t5)*alpha**2*K)
t1=r0**4*(gamma*(3*z**2+r0**2) + rho**2*(8*y**2-x**2))
t2=r0**2*(rho**4*(5*y**2+x**2) - 2*rho**2*z**2*(2*y**2+x**2)
-3*z**4*gamma)
t3=r**4*(2*y**4-gamma*(x**2+z**2))
t4=r0**2*( -gamma*(r0**2+2*z**2) - rho**2*(3*y**2-2*x**2) )
t5=r**2*( 2*y**4 - gamma*(x**2 + z**2) )
Byy=C*z/(2*alpha**4*beta**3*rho**4)* ( (t1-t2-t3)* E + (t4+t5)*alpha**2*K )
Bzz=C*z/(2*alpha**4*beta**3)* ( (6*r0**2*(rho**2-z**2) -7*r0**4 + r**4 )* E + (r0**2 - r**2)*alpha**2*K )
return (Bxx,Byy,Bzz)
def champB(self,x,z):
if abs(x) < 1e-8:
x=0
if x>0:
sx = 1
x = -x
else:
sx = -1
z = z-self.zs
x2 = x*x
z2 = z*z
r2 = x2+z2
b1 = self.a2+ r2
b2 = 2*x*self.a
b3 = b1+b2
b4 = b1-b2
b5 = -2*b2/b4
b6 = math.sqrt(b3/b4)*self.i
rb3 = math.sqrt(b3)
b7 = self.a*b3*rb3
b8 = self.a4-self.a2*(x2-2*z2)+z2*(x2+z2)
b9 = (self.a2+z2)*b3
e = ellipe(b5)
k = ellipk(b5)
bz = b6*((self.a2-r2)*e+b3*k)/b7
if x==0:
bx = 0.0
Atheta = 0.0
Adx = bz/2
else:
bx = -sx*z/x*b6*(b1*e-b3*k)/b7
Atheta = -sx*b6/x*(-b4*e+(self.a2+r2)*k)/(self.a*rb3)
Adx = b6/x2*(b8*e-b9*k)/b7
return [bx,bz,Atheta,Adx]
def champB(self, x, z):
if abs(x) < 1e-8:
x = 0
if x > 0:
sx = 1
x = -x
else:
sx = -1
z = z - self.zs
x2 = x * x
z2 = z * z
r2 = x2 + z2
b1 = self.a2 + r2
b2 = 2 * x * self.a
b3 = b1 + b2
b4 = b1 - b2
b5 = -2 * b2 / b4
b6 = math.sqrt(b3 / b4) * self.i
rb3 = math.sqrt(b3)
b7 = self.a * b3 * rb3
b8 = self.a4 - self.a2 * (x2 - 2 * z2) + z2 * (x2 + z2)
b9 = (self.a2 + z2) * b3
e = ellipe(b5)
k = ellipk(b5)
bz = b6 * ((self.a2 - r2) * e + b3 * k) / b7
if x == 0:
bx = 0.0
Atheta = 0.0
Adx = bz / 2
else:
bx = -sx * z / x * b6 * (b1 * e - b3 * k) / b7
Atheta = -sx * b6 / x * (-b4 * e +
(self.a2 + r2) * k) / (self.a * rb3)
Adx = b6 / x2 * (b8 * e - b9 * k) / b7
return [bx, bz, Atheta, Adx]
def s(ξ, ξ1=1e-5, ξ2=1e5):
r"""
Returns:
--------
.. math ::
2/\pi*E(ξ)+(ξ^2-1)*K(ξ) \text{ if } ξ<=1
2/\pi*ξ*E(1/ξ) \text{ if } ξ>1
Asymptotic cases are approximated by the asymptotic expressions.
"""
def B(x):
return ellipe(x ** 2) + (x ** 2 - 1) * ellipk(x ** 2)
m1 = ξ < ξ1
m4 = ξ > ξ2
m = ξ < 1
m2 = np.logical_and(m, np.logical_not(m1))
m3 = np.logical_and(np.logical_not(m), np.logical_not(m4))
r = np.zeros_like(ξ)
r[m1] = ξ[m1] ** 2 / 2
r[m4] = ξ[m4]
r[m2] = 2 / np.pi * B(ξ[m2])
r[m3] = 2 / np.pi * ξ[m3] * ellipe(1 / ξ[m3] ** 2)
return r
def greens(R, Z, Rc, Zc):
"""
Greens function for the toroidal elliptic operator
"""
ksq = 4.*R*Rc / ( (R + Rc)**2 + (Z - Zc)**2 ) # k**2
k = sqrt(ksq)
return sqrt(R*Rc) * ( (2. - ksq)*ellipk(k) - 2.*ellipe(k) ) / (2.*pi*k)
def WaveLengthDepth(k,a0,a1):
"""
Returns the wavelength and mean depth
of a cnoidal wave with parameters k,a0,a1
"""
kappa = np.sqrt(3*a1)/(2*np.sqrt(a0*(a0+a1)*(a0+(1-k*k)*a1)))
h0 = a0+ a1*special.ellipe(k)/special.ellipk(k)
lam = 2.*special.ellipk(k)/kappa
return lam,h0
def field(radius, z=0.0, r=0.0, pos=0.0):
"""Compute axial and radial component for a current loop
Based on formulas taken from
<http://www.netdenizen.com/emagnettest/offaxis/?offaxisloop>_
Parameters
----------
radius: float
Radius of the coil
z : float
The distance, on axis, from the center of the current loop
to the field measurement point.
r : float
The the radial distance from the axis of the current loop
to the field measurement point.
pos : float
The position of the center of the coil on the z axis
Returns
-------
Bz, Br : field in units of (Amperes * mu0 /2/radius)
"""
r = abs(r)
z = z - pos
alp = r / radius
bet = z / radius
Q = ((1 + alp) ** 2 + bet ** 2) # work on magnitude to improve performance
m = 4 * alp / Q
# K(k) is the complete elliptic integral function, of the first kind.
# E(k) is the complete elliptic integral function, of the second kind.
K = special.ellipkm1(1 - m)
E = special.ellipe(m)
# Bo is the magnetic field at the center of the coil (AMPERES MU0/2a)
# Bz the magnetic field component that is aligned with the coil axis and
# Br the magnetic field component that is in a radial direction.
Bz = Br = 0.0
if Q != 0.0 and Q - 4 * alp != 0.0 and z != pos:
Bz = (E * (1 - alp ** 2 - bet ** 2)
/ (Q - 4 * alp) + K) / pi / Q ** 0.5
if r > 0.000000 and Q != 0.0 and Q - 4 * alp != 0.0:
gam = z / r
Br = gam * (E * (1 + alp ** 2 + bet ** 2)
/ (Q - 4 * alp) - K) / pi / Q ** 0.5
return Bz, Br
def action_from_phase_amplitude(x2):
'''
Returns the relative action for given oscillation amplitude in time.
Action is normalised to the value at the separatrix, given in units of 1.
'''
action = np.zeros(len(x2))
indices = np.where(x2 != 1.)[0]
indices0 = np.where(x2 == 1.)[0]
action[indices] = (ellipe(x2[indices]) -
(1. - x2[indices])*ellipk(x2[indices]))
if indices0:
action[indices0] = np.float(ellipe(x2[indices0]))
return action
def G(x, t_cap, f, t):
r = 1. + f*math.cos(t)
rcap = 1 + x*math.cos(t_cap)
zmzcap = f*math.sin(t) - x*math.sin(t_cap)
k = 4*r*rcap/((r+rcap)**2 + zmzcap**2)
term1 = np.sqrt((r+rcap)**2 + zmzcap**2)
term1 = term1*rcap
term2 = (1.e0 - (k)*0.5e0)*scsp.ellipk(k)
term2 = term2 - scsp.ellipe(k)
return term1*term2
def BxForRot(self,xx,yy,zz): x = (xx-self.x0)*np.cos(self.theta) - (zz-self.z0)*np.sin(self.theta) y = yy-self.y0 z = (xx-self.x0)*np.sin(self.theta) + (zz-self.z0)*np.cos(self.theta) rho2 = x**2 + y**2 r2 = x**2 + y**2 + z**2 alpha2 = self.a**2 + r2 - 2*self.a*np.sqrt(rho2) beta2 = self.a**2 + r2 + 2*self.a*np.sqrt(rho2) k2 = 1 - alpha2/beta2 gamma = x**2 - y**2 C = mu0*self.I/np.pi return C*x*z/(2*alpha2*np.sqrt(beta2)*rho2 + epsilon)*\ ((self.a**2+r2)*sp.ellipe(k2)-alpha2*sp.ellipk(k2))*10**4
def _deriv(self, t, y, epsoct, phiq):
# Eqs. 11 of Katz (2011)
jz, Omega = y
CKL = phiq - jz**2 / 2.
x = (3 - 3 * CKL) / (3 + 2 * CKL)
fj = (15 * np.pi / (128 * np.sqrt(10)) / ellipk(x) * (4 - 11 * CKL)
* np.sqrt(6 + 4 * CKL))
fOmega = ((6 * ellipe(x) - 3 * ellipk(x)) / (4 * ellipk(x)))
jzdot = -epsoct * fj * np.sin(Omega)
Omegadot = jz * fOmega
return [jzdot, Omegadot]
def analyticalSolutionSolitary(x,t,a0,a1):
"""
Returns the cnoidal solution with parameters k,a0,a1
at (x,t) (possibly arrays)
"""
k = 0
g = 9.81
kappa = np.sqrt(3*a1)/(2*np.sqrt(a0*(a0+a1)))
h0 = a0+ a1*special.ellipe(k)/special.ellipk(k)
c = np.sqrt(g*a0*(a0+a1))
h = a0+ a1*np.power(np.cosh(kappa*(x-c*t)),-2)
u = c*(1-a0/h)
return h,u
def ellipse_circumf(major, minor, semi=True):
"""
Calculate the circumference of an ellipse given the axe lengths.
Parameters
----------
major : number
Major axis length
minor : number
Minor axis length
semi : Bool, default True
Use semi- (True) or full-axes (False) lengths
Returns
-------
area : number
Area of ellipse
"""
eccen2 = (major**2 - minor**2) / major**2
if semi:
return 4. * major * ellipe(eccen2)
elif not semi:
return 2. * major * ellipe(eccen2)
def _z2deriv(self,R,z,phi=0.,t=0.):
"""
NAME:
_z2deriv
PURPOSE:
evaluate the second vertical derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t- time
OUTPUT:
the second vertical derivative
HISTORY:
2018-08-04 - Written - Bovy (UofT)
"""
Raz2= (R+self.a)**2+z**2
m= 4.*R*self.a/Raz2
# Explicitly swapped in zforce here, so the z/z can be cancelled
# and z=0 is handled properly
return -4.*(3.*z**2/Raz2-1.
+4.*((1.+m)/(1.-m)-special.ellipk(m)/special.ellipe(m))\
*self.a*R*z**2/Raz2**2/m)\
*self.a/(1.-m)*((R+self.a)**2+z**2)**-1.5*special.ellipe(m)
def Bz(self,xx,yy,zz): """Return the z component of the magnetic field in the point of space (xx,yy,zz) """ x = xx-self.x0 y = yy-self.y0 z = zz-self.z0 rho2 = x**2 + y**2 r2 = x**2 + y**2 + z**2 alpha2 = self.a**2 + r2 - 2*self.a*np.sqrt(rho2) beta2 = self.a**2 + r2 + 2*self.a*np.sqrt(rho2) k2 = 1 - alpha2/beta2 gamma = x**2 - y**2 C = mu0*self.I/np.pi return C/(2*alpha2*np.sqrt(beta2))*\ ((self.a**2-r2)*sp.ellipe(k2)+alpha2*sp.ellipk(k2))*10**4
def analyticalSolution(x,t,k,a0,a1):
"""
Returns the cnoidal solution with parameters k,a0,a1
at (x,t) (possibly arrays)
"""
g = 9.81
kappa = np.sqrt(3*a1)/(2*np.sqrt(a0*(a0+a1)*(a0+(1-k*k)*a1)))
h0 = a0+ a1*special.ellipe(k)/special.ellipk(k)
c = np.sqrt(g*a0*(a0+a1)*(a0+(1.-k*k)*a1))/h0
sn,cn,dn,ph = special.ellipj(kappa*(x-c*t),k)
h = a0+a1*dn**2
u = c*(1-h0/h)
return h,u
def B(self,P):
a = self.a
theta = self.theta
x = P[0] - self.x0
y = P[1] - self.y0
r = x*np.cos(theta)+y*np.sin(theta)
z = -x*np.sin(theta)+y*np.cos(theta) # On se ramène à des
# coordonnées cylindriques par rapport à la spire. Pour la
# suite des calculs, voir l'aticle de T.Pré
# http://www.udppc.asso.fr/bupdoc/textes/fichierjoint/918/0918D119.zip
k= 4.*abs(r)*a/((a+abs(r))**2+z**2)
Kk=sp.ellipk(k)
Ek=sp.ellipe(k)
Br=self.I*(z/r)/np.sqrt((a+abs(r))**2+z**2)*(-Kk+(a**2+r**2+z**2)/((a-abs(r))**2+z**2)*Ek)
Bz=(self.I/np.sqrt((a+abs(r))**2+z**2))*(Kk+((a**2-r**2-z**2)/((a-abs(r))**2+z**2))*Ek)
return([Br*np.cos(theta)-Bz*np.sin(theta),Br*np.sin(theta)+Bz*np.cos(theta)])
def _vintersect_sphcyl_ellip(rs, rc, b):
"""
The cases for which evaluating the volume of intersection of a sphere with a cylinder
makes use of elliptic integrals.
"""
rs3 = rs ** 3
bprc = b + rc
bmrc = b - rc
A = max(rs ** 2, bprc ** 2)
B = min(rs ** 2, bprc ** 2)
C = bmrc ** 2
AB = A - B
AC = A - C
BC = B - C
k2 = BC / AC
s = bprc * bmrc
e1 = ellipk(k2)
e2 = ellipe(k2)
if bmrc == 0:
if rs == bprc:
vi = - 4. / 3 * AC ** 0.5 * (s + 2. / 3 * AC)
elif rs < bprc:
vi = 4. / 3 / A ** 0.5 * (e1 * AB * (3 * B - 2 * A) +
e2 * A * (2 * A - 4 * B)) / 3
else:
vi = 4. / 3 / A ** 0.5 * (e1 * AB * A -
e2 * A * (4 * A - 2 * B)) / 3
else:
a2 = 1 - B / C
e3 = elliptic_pi(a2, k2)
if rs == bprc:
vi = (4. / 3 * rs3 * np.arctan(2 * (b * rc) ** 0.5 / bmrc) -
4. / 3 * AC ** 0.5 * (s + 2. / 3 * AC))
elif rs < bprc:
vi = (4. / 3 / AC ** 0.5 *
(e3 * B ** 2 * s / C +
e1 * (s * (A - 2. * B) + AB * (3 * B - C - 2 * A) / 3) +
e2 * AC * (-s + (2 * A + 2 * C - 4 * B) / 3)))
else:
vi = 4. / 3 / AC ** 0.5 * (e3 * A ** 2 * s / C -
e1 * (A * s - AB * AC / 3.) -
e2 * AC * (s + (4. * A - 2 * B - 2 * C) / 3))
return vi
def force_between_windings(self, r1, r2, z, N_m):
m = 4*r1*r2/float((r1+r2)**2+z**2); # Argument of the Elliptic Integrals
K = special.ellipk(m); # Complete. First kind.
E = special.ellipe(m); # Complete. Second kind.
# Current in the coil.
I1 = self.c.I;
# The magnet is treated as an equivalent coil.
# I2 is the current in the equivalent coil.
I2 = self.m.B_r * self.m.l_m / float( N_m * MU_0);
return ( MU_0 * I1 * I2 * z
* np.sqrt(m/float(4*r1*r2))
* (K - E*(m/2.0-1)/float(m-1))
);
def heat_capacity(bond_energy, lower_temperature, higher_temperature, step=0.001):
"""
Calculate the exact heat capacity. Boltzmann constant is set to 1.
Formula from McCoy and Wu, 1973, The Two-Dimensional Ising Model.
"""
# Shorter variable name to ease formula writing.
j = bond_energy
exact_heat_capacity = []
for t in np.arange(lower_temperature, higher_temperature, step):
b = 1 / t
k = 2 * np.sinh(2 * b * j) / np.cosh(2 * b * j)**2
kprime = np.sqrt(1 - k**2)
c = (b * j / np.tanh(2 * b * j))**2 * (2 / np.pi) * (2 * ellipk(k**2) - 2 * ellipe(k**2) - (1 - kprime) * (np.pi / 2 + kprime * ellipk(k**2)))
exact_heat_capacity.append((t, c))
return exact_heat_capacity
def _zforce(self,R,z,phi=0.,t=0.):
"""
NAME:
_zforce
PURPOSE:
evaluate the vertical force for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the vertical force
HISTORY:
2018-08-04 - Written - Bovy (UofT)
"""
m= 4.*R*self.a/((R+self.a)**2+z**2)
return -4.*z*self.a/(1.-m)*((R+self.a)**2+z**2)**-1.5*special.ellipe(m)
def force_due_to_shell(self, r, z):
force = 0;
for e1 in [-1, 1]:
for e2 in [-1, 1]:
m1 = z - 0.5*e1*self.m.l_m - 0.5*e2*self.c.l_c;
m2 = ((self.m.r_m - r) / float(m1))**2 + 1;
m3 = np.sqrt((self.m.r_m + r)**2 + m1**2);
m4 = 4*self.m.r_m*r / float(m3**2); # Different from paper.
fs = ( special.ellipk(m4) - 1.0/m2*special.ellipe(m4)
+ (m1**2 / float(m3**2) - 1)
* mpmath.ellippi(m4/float(1-m2), m4));
force += e1*e2*m1*m2*m3*fs;
J1 = self.m.B_r;
J2 = MU_0*self.c.N_z*self.c.I/float(self.c.l_c);
return (J1*J2)/float(2*MU_0) * force;
def _show_currentloop_field():
'''http://www.netdenizen.com/emagnettest/offaxis/?offaxisloop
'''
from numpy import sqrt
r = numpy.linspace(0.0, 3.0, 51)
z = numpy.linspace(-1.0, 1.0, 51)
R, Z = numpy.meshgrid(r, z)
a = 1.0
V = 230 * sqrt(2.0)
rho = 1.535356e-08
II = V/rho
mu0 = pi * 4e-7
alpha = R / a
beta = Z / a
gamma = Z / R
Q = (1+alpha)**2 + beta**2
k = sqrt(4*alpha / Q)
from scipy.special import ellipk
from scipy.special import ellipe
Kk = ellipk(k**2)
Ek = ellipe(k**2)
B0 = mu0*II / (2*a)
V = B0 / (pi*sqrt(Q)) \
* (Ek * (1.0 - alpha**2 - beta**2)/(Q - 4*alpha) + Kk)
U = B0 * gamma / (pi*sqrt(Q)) \
* (Ek * (1.0 + alpha**2 + beta**2)/(Q - 4*alpha) - Kk)
Q = plt.quiver(R, Z, U, V)
plt.quiverkey(
Q, 0.7, 0.92, 1e4, '$1e4$',
labelpos='W',
fontproperties={'weight': 'bold'},
color='r'
)
plt.show()
return
def _Rforce(self,R,z,phi=0.,t=0.):
"""
NAME:
_Rforce
PURPOSE:
evaluate the radial force for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
the radial force
HISTORY:
2018-08-04 - Written - Bovy (UofT)
"""
m= 4.*R*self.a/((R+self.a)**2+z**2)
return -2.*self.a/R/nu.sqrt((R+self.a)**2+z**2)\
*(m*(R**2-self.a2-z**2)/4./(1.-m)/self.a/R*special.ellipe(m)
+special.ellipk(m))
def _Rzderiv(self,R,z,phi=0.,t=0.):
"""
NAME:
_Rzderiv
PURPOSE:
evaluate the mixed R,z derivative for this potential
INPUT:
R - Galactocentric cylindrical radius
z - vertical height
phi - azimuth
t - time
OUTPUT:
d2phi/dR/dz
HISTORY:
2018-08-04 - Written - Bovy (UofT)
"""
Raz2= (R+self.a)**2+z**2
m= 4.*R*self.a/Raz2
return (3.*(R+self.a)/Raz2
-2.*((1.+m)/(1.-m)-special.ellipk(m)/special.ellipe(m))\
*self.a*(self.a2+z**2-R**2)/Raz2**2/m)*self._zforce(R,z)
def create_mesh(axis0=1, axis1=0.5, num_boundary_points=100):
# lengths of major and minor axes
a = max(axis0, axis1)
b = min(axis0, axis1)
# Choose the maximum area of a triangle equal to the area of
# an equilateral triangle on the boundary.
# For circumference of an ellipse, see
# http://en.wikipedia.org/wiki/Ellipse#Circumference
eccentricity = np.sqrt(1.0 - (b / a) ** 2)
length_boundary = float(4 * a * special.ellipe(eccentricity))
a_boundary = length_boundary / num_boundary_points
max_area = a_boundary ** 2 * np.sqrt(3) / 4
# generate points on the circle
Phi = np.linspace(0, 2 * np.pi, num_boundary_points, endpoint=False)
boundary_points = np.column_stack((a * np.cos(Phi), b * np.sin(Phi)))
info = meshpy.triangle.MeshInfo()
info.set_points(boundary_points)
def _round_trip_connect(start, end):
result = []
for i in range(start, end):
result.append((i, i + 1))
result.append((end, start))
return result
info.set_facets(_round_trip_connect(0, len(boundary_points) - 1))
def _needs_refinement(vertices, area):
return bool(area > max_area)
meshpy_mesh = meshpy.triangle.build(info, refinement_func=_needs_refinement)
# append column
pts = np.array(meshpy_mesh.points)
points = np.c_[pts[:, 0], pts[:, 1], np.zeros(len(pts))]
return points, np.array(meshpy_mesh.elements)
def computeMajor(ratio, circum):
"""
Use the complete elliptic integrals of second kind to figure out what value
of major axis corresponds to the circumference, given b/a ratio
:math:`circum = 4 a E(k)` where :math:`k = \\sqrt{ 1- b^2/a^2 } = \\sqrt{1 - ratio^2}`
therefore, :math:`a = circum/(4E(k))`
*Args:*
ratio:
ratio of semi-minor to semi-major axis of Ellipse, ratio = b/a <= 1
circum:
circumference/perimeter of ellipse
*Returns:*
a:
semi-major axis of ellipse
"""
k = np.sqrt(1- ratio**2)
a = circum/(4*scsp.ellipe(k))
return a