def __str__(self):
thestr = "0"
var = self.variable
# Remove leading zeros
coeffs = self.coeffs[NX.logical_or.accumulate(self.coeffs != 0)]
N = len(coeffs) - 1
def fmt_float(q):
s = '%.4g' % q
if s.endswith('.0000'):
s = s[:-5]
return s
for k in range(len(coeffs)):
if not iscomplex(coeffs[k]):
coefstr = fmt_float(real(coeffs[k]))
elif real(coeffs[k]) == 0:
coefstr = '%sj' % fmt_float(imag(coeffs[k]))
else:
coefstr = '(%s + %sj)' % (fmt_float(real(
coeffs[k])), fmt_float(imag(coeffs[k])))
power = (N - k)
if power == 0:
if coefstr != '0':
newstr = '%s' % (coefstr, )
else:
if k == 0:
newstr = '0'
else:
newstr = ''
elif power == 1:
if coefstr == '0':
newstr = ''
elif coefstr == 'b':
newstr = var
else:
newstr = '%s %s' % (coefstr, var)
else:
if coefstr == '0':
newstr = ''
elif coefstr == 'b':
newstr = '%s**%d' % (
var,
power,
)
else:
newstr = '%s %s**%d' % (coefstr, var, power)
if k > 0:
if newstr != '':
if newstr.startswith('-'):
thestr = "%s - %s" % (thestr, newstr[1:])
else:
thestr = "%s + %s" % (thestr, newstr)
else:
thestr = newstr
return _raise_power(thestr)
def __str__(self):
thestr = "0"
var = self.variable
# Remove leading zeros
coeffs = self.coeffs[NX.logical_or.accumulate(self.coeffs != 0)]
N = len(coeffs) - 1
def fmt_float(q):
s = "%.4g" % q
if s.endswith(".0000"):
s = s[:-5]
return s
for k in range(len(coeffs)):
if not iscomplex(coeffs[k]):
coefstr = fmt_float(real(coeffs[k]))
elif real(coeffs[k]) == 0:
coefstr = "%sj" % fmt_float(imag(coeffs[k]))
else:
coefstr = "(%s + %sj)" % (
fmt_float(real(coeffs[k])),
fmt_float(imag(coeffs[k])),
)
power = N - k
if power == 0:
if coefstr != "0":
newstr = "%s" % (coefstr,)
else:
if k == 0:
newstr = "0"
else:
newstr = ""
elif power == 1:
if coefstr == "0":
newstr = ""
elif coefstr == "b":
newstr = var
else:
newstr = "%s %s" % (coefstr, var)
else:
if coefstr == "0":
newstr = ""
elif coefstr == "b":
newstr = "%s**%d" % (var, power)
else:
newstr = "%s %s**%d" % (coefstr, var, power)
if k > 0:
if newstr != "":
if newstr.startswith("-"):
thestr = "%s - %s" % (thestr, newstr[1:])
else:
thestr = "%s + %s" % (thestr, newstr)
else:
thestr = newstr
return _raise_power(thestr)
def __str__(self):
thestr = "0"
var = self.variable
# Remove leading zeros
coeffs = self.coeffs[NX.logical_or.accumulate(self.coeffs != 0)]
N = len(coeffs)-1
def fmt_float(q):
s = '%.4g' % q
if s.endswith('.0000'):
s = s[:-5]
return s
for k in range(len(coeffs)):
if not iscomplex(coeffs[k]):
coefstr = fmt_float(real(coeffs[k]))
elif real(coeffs[k]) == 0:
coefstr = '%sj' % fmt_float(imag(coeffs[k]))
else:
coefstr = '(%s + %sj)' % (fmt_float(real(coeffs[k])),
fmt_float(imag(coeffs[k])))
power = (N-k)
if power == 0:
if coefstr != '0':
newstr = '%s' % (coefstr,)
else:
if k == 0:
newstr = '0'
else:
newstr = ''
elif power == 1:
if coefstr == '0':
newstr = ''
elif coefstr == 'b':
newstr = var
else:
newstr = '%s %s' % (coefstr, var)
else:
if coefstr == '0':
newstr = ''
elif coefstr == 'b':
newstr = '%s**%d' % (var, power,)
else:
newstr = '%s %s**%d' % (coefstr, var, power)
if k > 0:
if newstr != '':
if newstr.startswith('-'):
thestr = "%s - %s" % (thestr, newstr[1:])
else:
thestr = "%s + %s" % (thestr, newstr)
else:
thestr = newstr
return _raise_power(thestr)
def __str__(self):
thestr = "0"
var = self.variable
# Remove leading zeros
coeffs = self.coeffs[NX.logical_or.accumulate(self.coeffs != 0)]
N = len(coeffs) - 1
def fmt_float(q):
s = "%.4g" % q
if s.endswith(".0000"):
s = s[:-5]
return s
for k in range(len(coeffs)):
if not iscomplex(coeffs[k]):
coefstr = fmt_float(real(coeffs[k]))
elif real(coeffs[k]) == 0:
coefstr = "%sj" % fmt_float(imag(coeffs[k]))
else:
coefstr = "(%s + %sj)" % (fmt_float(real(coeffs[k])), fmt_float(imag(coeffs[k])))
power = N - k
if power == 0:
if coefstr != "0":
newstr = "%s" % (coefstr,)
else:
if k == 0:
newstr = "0"
else:
newstr = ""
elif power == 1:
if coefstr == "0":
newstr = ""
elif coefstr == "b":
newstr = var
else:
newstr = "%s %s" % (coefstr, var)
else:
if coefstr == "0":
newstr = ""
elif coefstr == "b":
newstr = "%s**%d" % (var, power)
else:
newstr = "%s %s**%d" % (coefstr, var, power)
if k > 0:
if newstr != "":
if newstr.startswith("-"):
thestr = "%s - %s" % (thestr, newstr[1:])
else:
thestr = "%s + %s" % (thestr, newstr)
else:
thestr = newstr
return _raise_power(thestr)
def make_poly(facts):
""" Create polynomial from factors """
poly = [1]
for factor in facts:
poly = polymul(poly, [1, -factor])
return real(poly)
def make_poly(facts):
""" Create polynomial from factors """
poly = [1]
for factor in facts:
poly = polymul(poly, [1, -factor])
return real(poly)
def GetTimeFlow(self, el, time):
'''
Calculating inlet flow (coefficients of the FFT x(t)=A0+sum(2*Ck*exp(j*k*2*pi*f*t)))
for a specific time value.
If signal is specified, flow is computed from time values.
'''
try:
period = self.SimulationContext.Context['period']
except KeyError:
print "Error, Please set period in Simulation Context XML File"
raise
try:
signal = self.InFlows[el]['signal']
try:
timestep = self.SimulationContext.Context['timestep']
except KeyError:
print "Error, Please set timestep in Simulation Context XML File"
raise
t = arange(0.0,period+timestep,timestep)
t2 = list(t)
Flow = float(signal[t2.index(time)])/6.0e7
self.Flow = Flow
return Flow
except KeyError:
f_coeff = self.InFlows[el]['f_coeff']
A0 = self.InFlows[el]['A0']
Cc = f_coeff*1.0/2.0*1e-6
Flow = A0
for k in arange(0,f_coeff.shape[0]):
Flow += real(2.0*complex(Cc[k,0],Cc[k,1])*exp(1j*(k+1)*2.0*pi*time/period))
self.Flow = Flow
return Flow
def GetFlow(self):
'''
Calculating inlet flow (coefficients of the FFT x(t)=A0+sum(2*Ck*exp(j*k*2*pi*f*t)))
Timestep and period from SimulationContext are necessary.
'''
try:
timestep = self.SimulationContext.Context['timestep']
except KeyError:
print "Error, Please set timestep in Simulation Context XML File"
raise
try:
period = self.SimulationContext.Context['period']
except KeyError:
print "Error, Please set period in Simulation Context XML File"
raise
t = arange(0.0,period+timestep,timestep).reshape((1,ceil(period/timestep+1.0)))
Cc = self.f_coeff*1.0/2.0*1e-6
Flow = zeros((1, ceil(period/timestep+1.0)))
for freq in arange(0,ceil(period/timestep+1.0)):
Flow[0, freq] = self.A0_v
for k in arange(0,self.f_coeff.shape[0]):
Flow[0, freq] = Flow[0, freq]+real(2.0*complex(Cc[k,0],Cc[k,1])*exp(1j*(k+1)*2.0*pi*t[0,freq]/period))
self.Flow = Flow
return Flow
def transform(self, R, s, t, pos, ori=nan):
""" Transform node position. """
assert None not in (R, s, t)
assert not imag(R).any()
R = real(R)
if not isnan(ori):
# angle of rotation matrix R in ccw
Rtheta = arctan2(R[1, 0], R[0, 0])
ori = mod(ori - Rtheta, 2 * pi)
return concatenate((t + dot(dot(s, R), pos), [ori]))
def transform(self, R, s, t, pos, ori=nan):
""" Transform node position. """
assert None not in (R, s, t)
assert not imag(R).any()
R = real(R)
if not isnan(ori):
# angle of rotation matrix R in ccw
Rtheta = arctan2(R[1, 0], R[0, 0])
ori = mod(ori - Rtheta, 2*pi)
return concatenate((t + dot(dot(s, R), pos), [ori]))
def TheoGarrAerofoil(alphaBar,betaBar,hBar,phi0,phi1,phi2,a,e,k,nT):
"""@brief Lift and drag of thin aerofoil oscillating in pitch, plunge and
flap angle.
@param alphaBar amplitude of pitching oscillations.
@param betaBar amplitude of flap oscillations.
@param hBar amplitude of plunging oscillations.
@param phi0 Phase angle in pitch.
@param phi1 Phase angle in flap angle.
@param phi2 Phase angle in plunge.
@param a Non-dim offset of pitching axis aft of mid-chord.
@param e Non-dim offset of flap hinge aft of mid-chord.
@param k Reduced frequency, k = omega*b/U.
@param nT Number of periods in output.
@details As this will likely be extended to also calculate moments there are
many commented variables which may be of use later on - don't remove them."""
# Time parameters.
sFinal = 2 * pi * nT / k
s = np.linspace(0.0, sFinal, nT * 100, True)
# Kinematic Parameters
# Sinusoidal variations, take imaginary part of Theodorsen's equations.
alpha = alphaBar * sin(k*s + phi0)
alphaDot = k * alphaBar * cos(k*s + phi0)
alphaDotDot = - pow(k,2.0) * alphaBar * sin(k*s + phi0)
# h = hBar * sin(k*s + phi1)
hDot = k * hBar * cos(k*s + phi1)
hDotDot = - pow(k,2.0) * hBar * sin(k*s + phi1)
beta = betaBar * sin(k*s + phi2)
betaDot = k * betaBar * cos(k*s + phi2)
betaDotDot = - pow(k,2.0) * betaBar * sin(k*s + phi2)
# Theodorsen's constants
F1 = e * arccos(e) - 1.0/3.0 * (2 + pow(e, 2.0)) * np.sqrt(1 - pow(e, 2.0))
F2 = (e * (1 - pow(e, 2.0)) -
(1 + pow(e, 2.0)) * np.sqrt(1 - pow(e, 2.0)) * arccos(e) +
e*pow(arccos(e), 2.0))
# F3 = (- (1.0/8.0 + pow(e, 2.0)) * pow(arccos(e),2.0) +
# 1.0/4.0 * e * np.sqrt(1 - pow(e, 2.0)) * arccos(e)
# * (7 + 2*pow(e, 2.0))
# - 1.0/8.0 * (1 - pow(e, 2.0)) * (5 * pow(e, 2.0) + 4))
F4 = e * np.sqrt(1 - pow(e, 2.0)) - arccos(e)
F5 = (-(1 - pow(e, 2.0)) - pow(arccos(e),2.0)
+ 2 * e * np.sqrt(1 - pow(e, 2.0)) * arccos(e))
# F6 = F2
# F7 = (-(1.0/8.0 + pow(e, 2.0)) * arccos(e)
# + 1.0/8.0 * e * np.sqrt(1 - pow(e, 2.0)) * (7 + 2 * pow(e, 2.0)))
# F8 = (-1.0/3.0 * np.sqrt(1 - pow(e, 2.0)) * (2 * pow(e, 2.0) + 1)
# + e * arccos(e))
F9 = 1.0/2.0 * (1.0/3.0 * pow(np.sqrt(1 - pow(e, 2.0)),3) + a * F4)
F10 = np.sqrt(1 - pow(e, 2.0)) + arccos(e)
F11 = (1 - 2 * e) * arccos(e) + (2 - e) * np.sqrt(1 - pow(e, 2.0))
# F12 = np.sqrt(1 - pow(e, 2.0)) * (2 + e) - arccos(e) * (2 * e + 1)
# F13 = 1.0/2.0 * (-F7 - (e - a) * F1)
# F14 = -arccos(e) + e*np.sqrt(1-pow(e, 2.0))
# F14 = 1.0/16.0 + 1.0/2.0 * a * e
# F6 = F2
# Garrick's constants
# F15 = F4 + F10
# F16 = F1 - F8 - (e - a) * F4 + 1.0/2.0 * F11
# F17 = -2 * F9 - F1 + (a - 0.5) * F4
# F18 = F5 - F4 * F10
# F19 = F4 * F11
F20 = -np.sqrt(1 - pow(e, 2.0)) + arccos(e)
M = ( 2 * real(C(k)) * (alphaBar * cos(phi0)
- hBar * k * sin(phi2)
- (0.5 - a) * alphaBar * k * sin(phi0)
+ F10/pi * betaBar * cos(phi1)
- F11/(2*pi) * betaBar * k * sin(phi1))
- 2 * imag(C(k)) * (alphaBar * sin(phi0)
+ hBar * k * cos(phi2)
+ (0.5 - a) * alphaBar * k * cos(phi0)
+ F10/pi * betaBar * sin(phi1)
+ F11/(2*pi) * betaBar * k * cos(phi1))
+ alphaBar * k * sin(phi0)
- 2.0/pi * np.sqrt(1 - pow(e, 2.0)) * betaBar * cos(phi1)
- F4/pi * betaBar * k * sin(phi1) )
N = ( 2 * real(C(k)) * (alphaBar * sin(phi0)
+ hBar * k * cos(phi2)
+ (0.5 - a) * alphaBar * k * cos(phi0)
+ F10/pi * betaBar * sin(phi1)
+ F11/(2*pi) * betaBar * k * cos(phi1) )
+ 2 * imag(C(k)) * (alphaBar * cos(phi0)
- hBar * k * sin(phi2)
- (0.5 - a) * alphaBar * k * sin(phi0)
+ F10/pi * betaBar * cos(phi1)
- F11/(2*pi) * betaBar * k * sin(phi1))
- alphaBar * k * cos(phi0)
- 2.0/pi * np.sqrt(1 - pow(e, 2.0)) * betaBar * sin(phi1)
- F4/pi * betaBar * k * cos(phi1) )
# Loop through time steps
p = ( -(pi*alphaDot + pi*hDotDot - pi*a*alphaDotDot
- F4*betaDot - F1*betaDotDot)
- 2*pi*real(C(k)) * (alpha + hDot + (0.5 - a)*alphaDot + F10/pi*beta
+ F11/(2*pi)*betaDot)
- 2*pi*imag(C(k))/k * (alphaDot + hDotDot + alphaDotDot
+ F10/pi*betaDot + F11/(2*pi)*betaDotDot) )
pBeta = ( -(-F4*alphaDot - F4*hDotDot + F9*alphaDotDot - F5/(2*pi)*betaDot
- F2/(2*pi)*betaDotDot)
- 2*np.sqrt(1 - pow(e,2.0)) * (0.5*(1 - e)*alphaDot
+ np.sqrt(1 - pow(e,2.0))/pi*beta
+ F10/(2*pi)*(1 - e)*betaDot)
- 2*F20*real(C(k))*(alpha + hDot + (0.5 - a)*alphaDot
+ F10/pi*beta + F11/(2*pi)*betaDot)
- 2*F20*imag(C(k))/k*(alphaDot + hDotDot
+ (0.5 - a)*alphaDotDot
+ F10/pi*betaDot
+ F11/(2*pi)*betaDotDot) )
S = (np.sqrt(2.0) / 2.0) * (M * sin(k * s) +
N * cos(k * s) )
d = - alpha*p - pi*pow(S, 2.0) - beta*pBeta
Cl = - p
Cd = d
return Cl, Cd, alpha, beta, hDot, s
d = - alpha*p - pi*pow(S, 2.0) - beta*pBeta
Cl = - p
Cd = d
return Cl, Cd, alpha, beta, hDot, s
if __name__ == '__main__':
# Test Theodorsen function.
if False:
k = np.linspace(0,1,101,True)
Ck = np.zeros_like(k,complex)
for kIter in range(k.shape[0]):
Ck[kIter] = C(k[kIter])
plt.plot(k,real(Ck),k,imag(Ck))
plt.grid()
plt.xlabel('k')
plt.legend(['Re(C(k))','Im(C(k))'])
plt.show()
# Test TheoGarr.
if True:
alphaBar = 0.0*pi/180.0
betaBar = 0.1*pi/180.0
hBar = 0.0
phi0 =0.0*pi/180.0
phi1 = 0.0*pi/180.0
phi2 = 0.0*pi/180.0
a = -0.5
e = 0.6
def TheoGarrAerofoil(alphaBar, betaBar, hBar, phi0, phi1, phi2, a, e, k, nT):
"""@brief Lift and drag of thin aerofoil oscillating in pitch, plunge and
flap angle.
@param alphaBar amplitude of pitching oscillations.
@param betaBar amplitude of flap oscillations.
@param hBar amplitude of plunging oscillations.
@param phi0 Phase angle in pitch.
@param phi1 Phase angle in flap angle.
@param phi2 Phase angle in plunge.
@param a Non-dim offset of pitching axis aft of mid-chord.
@param e Non-dim offset of flap hinge aft of mid-chord.
@param k Reduced frequency, k = omega*b/U.
@param nT Number of periods in output.
@details As this will likely be extended to also calculate moments there are
many commented variables which may be of use later on - don't remove them."""
# Time parameters.
sFinal = 2 * pi * nT / k
s = np.linspace(0.0, sFinal, nT * 100, True)
# Kinematic Parameters
# Sinusoidal variations, take imaginary part of Theodorsen's equations.
alpha = alphaBar * sin(k * s + phi0)
alphaDot = k * alphaBar * cos(k * s + phi0)
alphaDotDot = -pow(k, 2.0) * alphaBar * sin(k * s + phi0)
# h = hBar * sin(k*s + phi1)
hDot = k * hBar * cos(k * s + phi1)
hDotDot = -pow(k, 2.0) * hBar * sin(k * s + phi1)
beta = betaBar * sin(k * s + phi2)
betaDot = k * betaBar * cos(k * s + phi2)
betaDotDot = -pow(k, 2.0) * betaBar * sin(k * s + phi2)
# Theodorsen's constants
F1 = e * arccos(e) - 1.0 / 3.0 * (2 + pow(e, 2.0)) * np.sqrt(1 -
pow(e, 2.0))
F2 = (e * (1 - pow(e, 2.0)) -
(1 + pow(e, 2.0)) * np.sqrt(1 - pow(e, 2.0)) * arccos(e) +
e * pow(arccos(e), 2.0))
# F3 = (- (1.0/8.0 + pow(e, 2.0)) * pow(arccos(e),2.0) +
# 1.0/4.0 * e * np.sqrt(1 - pow(e, 2.0)) * arccos(e)
# * (7 + 2*pow(e, 2.0))
# - 1.0/8.0 * (1 - pow(e, 2.0)) * (5 * pow(e, 2.0) + 4))
F4 = e * np.sqrt(1 - pow(e, 2.0)) - arccos(e)
F5 = (-(1 - pow(e, 2.0)) - pow(arccos(e), 2.0) +
2 * e * np.sqrt(1 - pow(e, 2.0)) * arccos(e))
# F6 = F2
# F7 = (-(1.0/8.0 + pow(e, 2.0)) * arccos(e)
# + 1.0/8.0 * e * np.sqrt(1 - pow(e, 2.0)) * (7 + 2 * pow(e, 2.0)))
# F8 = (-1.0/3.0 * np.sqrt(1 - pow(e, 2.0)) * (2 * pow(e, 2.0) + 1)
# + e * arccos(e))
F9 = 1.0 / 2.0 * (1.0 / 3.0 * pow(np.sqrt(1 - pow(e, 2.0)), 3) + a * F4)
F10 = np.sqrt(1 - pow(e, 2.0)) + arccos(e)
F11 = (1 - 2 * e) * arccos(e) + (2 - e) * np.sqrt(1 - pow(e, 2.0))
# F12 = np.sqrt(1 - pow(e, 2.0)) * (2 + e) - arccos(e) * (2 * e + 1)
# F13 = 1.0/2.0 * (-F7 - (e - a) * F1)
# F14 = -arccos(e) + e*np.sqrt(1-pow(e, 2.0))
# F14 = 1.0/16.0 + 1.0/2.0 * a * e
# F6 = F2
# Garrick's constants
# F15 = F4 + F10
# F16 = F1 - F8 - (e - a) * F4 + 1.0/2.0 * F11
# F17 = -2 * F9 - F1 + (a - 0.5) * F4
# F18 = F5 - F4 * F10
# F19 = F4 * F11
F20 = -np.sqrt(1 - pow(e, 2.0)) + arccos(e)
M = (2 * real(C(k)) * (alphaBar * cos(phi0) - hBar * k * sin(phi2) -
(0.5 - a) * alphaBar * k * sin(phi0) +
F10 / pi * betaBar * cos(phi1) - F11 /
(2 * pi) * betaBar * k * sin(phi1)) -
2 * imag(C(k)) * (alphaBar * sin(phi0) + hBar * k * cos(phi2) +
(0.5 - a) * alphaBar * k * cos(phi0) +
F10 / pi * betaBar * sin(phi1) + F11 /
(2 * pi) * betaBar * k * cos(phi1)) +
alphaBar * k * sin(phi0) -
2.0 / pi * np.sqrt(1 - pow(e, 2.0)) * betaBar * cos(phi1) -
F4 / pi * betaBar * k * sin(phi1))
N = (2 * real(C(k)) * (alphaBar * sin(phi0) + hBar * k * cos(phi2) +
(0.5 - a) * alphaBar * k * cos(phi0) +
F10 / pi * betaBar * sin(phi1) + F11 /
(2 * pi) * betaBar * k * cos(phi1)) +
2 * imag(C(k)) * (alphaBar * cos(phi0) - hBar * k * sin(phi2) -
(0.5 - a) * alphaBar * k * sin(phi0) +
F10 / pi * betaBar * cos(phi1) - F11 /
(2 * pi) * betaBar * k * sin(phi1)) -
alphaBar * k * cos(phi0) -
2.0 / pi * np.sqrt(1 - pow(e, 2.0)) * betaBar * sin(phi1) -
F4 / pi * betaBar * k * cos(phi1))
# Loop through time steps
p = (-(pi * alphaDot + pi * hDotDot - pi * a * alphaDotDot - F4 * betaDot -
F1 * betaDotDot) - 2 * pi * real(C(k)) *
(alpha + hDot + (0.5 - a) * alphaDot + F10 / pi * beta + F11 /
(2 * pi) * betaDot) - 2 * pi * imag(C(k)) / k *
(alphaDot + hDotDot + alphaDotDot + F10 / pi * betaDot + F11 /
(2 * pi) * betaDotDot))
pBeta = (-(-F4 * alphaDot - F4 * hDotDot + F9 * alphaDotDot - F5 /
(2 * pi) * betaDot - F2 /
(2 * pi) * betaDotDot) - 2 * np.sqrt(1 - pow(e, 2.0)) *
(0.5 *
(1 - e) * alphaDot + np.sqrt(1 - pow(e, 2.0)) / pi * beta + F10 /
(2 * pi) * (1 - e) * betaDot) - 2 * F20 * real(C(k)) *
(alpha + hDot + (0.5 - a) * alphaDot + F10 / pi * beta + F11 /
(2 * pi) * betaDot) - 2 * F20 * imag(C(k)) / k *
(alphaDot + hDotDot +
(0.5 - a) * alphaDotDot + F10 / pi * betaDot + F11 /
(2 * pi) * betaDotDot))
S = (np.sqrt(2.0) / 2.0) * (M * sin(k * s) + N * cos(k * s))
d = -alpha * p - pi * pow(S, 2.0) - beta * pBeta
Cl = -p
Cd = d
return Cl, Cd, alpha, beta, hDot, s
d = -alpha * p - pi * pow(S, 2.0) - beta * pBeta
Cl = -p
Cd = d
return Cl, Cd, alpha, beta, hDot, s
if __name__ == '__main__':
# Test Theodorsen function.
if False:
k = np.linspace(0, 1, 101, True)
Ck = np.zeros_like(k, complex)
for kIter in range(k.shape[0]):
Ck[kIter] = C(k[kIter])
plt.plot(k, real(Ck), k, imag(Ck))
plt.grid()
plt.xlabel('k')
plt.legend(['Re(C(k))', 'Im(C(k))'])
plt.show()
# Test TheoGarr.
if True:
alphaBar = 0.0 * pi / 180.0
betaBar = 0.1 * pi / 180.0
hBar = 0.0
phi0 = 0.0 * pi / 180.0
phi1 = 0.0 * pi / 180.0
phi2 = 0.0 * pi / 180.0
a = -0.5
e = 0.6