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CerfonFreidberg.py
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CerfonFreidberg.py
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# Compute poloidal flux function Psi from analytic Grad-Shafranov solutions in
# Cerfon & Freidberg, Physics of Plasmas 17, 032502 (2010); doi: 10.1063/1.3328818
# Matlab implementation of solutions for c1..c12 by James Cook <[email protected]> (2013)
# Python implementation by John Omotani <[email protected]> (2015)
import numpy
import scipy.optimize
import scipy.integrate
import sympy
from matplotlib import pyplot
import math
class CerfonFreidberg:
"""
x and y are normalised coordinates, x=R/R0 and y=Z/R0
epsi is the inverse aspect ratio
kapp is the elongation
delt is the triangularity
xsep, ysep are the coordinates of the X-point (ysep<0 by assumption)
qsta is 'q_*' as defined in the Cerfon & Freidberg paper; it is not actually used here
A is a parameter whose value determines the toroidal beta of the equilibrium (see C&F)
R0 is the nominal major radius of the plasma
B0 is the vacuum magnetic field at R0
"""
integrationGridSize = 1000
def initByName(self,machineName):
"""
Initialize using parameters representative of machines:
"ITER": See R Aymar, Barabaschi and Shimomura 2002 Plasma Phys. Control. Fusion 44 519
"NSTX": ??? Ono, Masayuki, S. M. Kaye, Y-KM Peng, G. Barnes, W. Blanchard, M. D. Carter, J. Chrzanowski et al. "Exploration of spherical torus physics in the NSTX device." Nuclear Fusion 40, no. 3Y (2000): 557.
"""
if machineName == "ITER":
epsi = 0.32
kapp = 1.7
delt = 0.33
xsep = 0.88
ysep = -0.6
qsta = 1.57
#A = -0.155 # C&F's value
A = -0.0825542587266
R0 = 6.2
B0 = 5.3
#Psi0 = 228.820949438 # from betat using C&F's A
Psi0 = 236.265277402
# plotting stuff
xmin = 0.5
xmax = 1.5
ymin = -0.8
ymax = 0.6
elif machineName == "NSTX":
epsi = 0.78
kapp = 2
delt = 0.35
xsep = 0.7
ysep = -1.71
qsta = 2
A = -0.05
R0 = 0.85
B0 = 0.3
Psi0 = None
# plotting stuff
xmin = 0
xmax = 2
ymin = -2
ymax = 1.6
else:
print("Unknown machineName:",machineName)
raise Exception
self.init(epsi,kapp,delt,xsep,ysep,A,R0,B0,Psi0)
def init(self,epsi,kapp,delt,xsep,ysep,A,R0,B0,Psi0=None):
"""
Initialize by giving desired parameter values. Psi0 can be calculated from other quantities, e.g. toroidal beta, if not already known
"""
self.epsi = epsi
self.kapp = kapp
self.delt = delt
self.xsep = xsep
self.ysep = ysep
self.A = A
self.R0 = R0
self.B0 = B0
self.Psi0 = Psi0 # call calculatePsi0FromBetat or calculatePsi0FromCurrent to get this value, if needed
self.mu0 = 4.e-7*numpy.pi
self.Cp = None # call calculateCp to get this value, if needed
self.xmin = .8*(1.-self.epsi)
self.xmax = 1.+self.epsi+.2*(1.-self.epsi)
self.ymin = 1.2*self.ysep
self.ymax = 1.2*self.epsi*self.kapp
if A is not None:
# Can only actually solve for coefficients if A is given, otherwise calculateAAndPsi0FromBetatAndCurrent() must be called to calculate A
alph = numpy.arcsin(delt)
N1 = -(1.+alph)**2/epsi/kapp**2
N2 = (1.-alph)**2/epsi/kapp**2
N3 = -kapp/epsi/numpy.cos(alph)**2
x,y = sympy.symbols("x y")
c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12 = sympy.symbols("c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12")
psi = CerfonFreidberg._getpsi(x,y,A,c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12)
psi_x = sympy.diff(psi,x)
psi_xx = sympy.diff(psi_x,x)
psi_y = sympy.diff(psi,y)
psi_yy = sympy.diff(psi_y,y)
# solve for coefficients
s = sympy.solvers.solve( [psi.subs([(x,1.+epsi),(y,0.)]),
psi.subs([(x,1.-epsi),(y,0.)]),
psi.subs([(x,1.-delt*epsi),(y,kapp*epsi)]),
psi.subs([(x,xsep),(y,ysep)]),
psi_y.subs([(x,1.+epsi),(y,0.)]),
psi_y.subs([(x,1.-epsi),(y,0.)]),
psi_x.subs([(x,1.-delt*epsi),(y,kapp*epsi)]),
psi_x.subs([(x,xsep),(y,ysep)]),
psi_y.subs([(x,xsep),(y,ysep)]),
(psi_yy+N1*psi_x).subs([(x,1.+epsi),(y,0.)]),
(psi_yy+N2*psi_x).subs([(x,1.-epsi),(y,0.)]),
(psi_xx+N3*psi_y).subs([(x,1.-delt*epsi),(y,kapp*epsi)])
]
,[c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12]
)
# evaluate coefficients.
self.c1 = s[c1]
self.c2 = s[c2]
self.c3 = s[c3]
self.c4 = s[c4]
self.c5 = s[c5]
self.c6 = s[c6]
self.c7 = s[c7]
self.c8 = s[c8]
self.c9 = s[c9]
self.c10 = s[c10]
self.c11 = s[c11]
self.c12 = s[c12]
self.Raxis = None
self.Zaxis = None
self.Psiaxis = None
if Psi0:
# Psiaxis only makes sense if the value of Psi0 is known
self.getAxis() # Set the values of Raxis, Zaxis, Psiaxis
#print("c1",c1)
#print("c2",c2)
#print("c3",c3)
#print("c4",c4)
#print("c5",c5)
#print("c6",c6)
#print("c7",c7)
#print("c8",c8)
#print("c9",c9)
#print("c10",c10)
#print("c11",c11)
#print("c12",c12)
#print("test1",CerfonFreidberg._getpsi(1.+epsi,0.,A,c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12))
#print("test2",CerfonFreidberg._getpsi(1.-epsi,0.,A,c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12))
#print("test3",CerfonFreidberg._getpsi(1.-delt*epsi,kapp*epsi,A,c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12))
#print("test4",CerfonFreidberg._getpsi(xsep,ysep,A,c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12))
#print("test5",psi_y.subs([
# (x,1.+epsi),
# (y,0.),
# (sympy.Symbol('c1'),c1),
# (sympy.Symbol('c2'),c2),
# (sympy.Symbol('c3'),c3),
# (sympy.Symbol('c4'),c4),
# (sympy.Symbol('c5'),c5),
# (sympy.Symbol('c6'),c6),
# (sympy.Symbol('c7'),c7),
# (sympy.Symbol('c8'),c8),
# (sympy.Symbol('c9'),c9),
# (sympy.Symbol('c10'),c10),
# (sympy.Symbol('c11'),c11),
# (sympy.Symbol('c12'),c12)]))
#print("test6",psi_y.subs([
# (x,1.-epsi),
# (y,0.),
# (sympy.Symbol('c1'),c1),
# (sympy.Symbol('c2'),c2),
# (sympy.Symbol('c3'),c3),
# (sympy.Symbol('c4'),c4),
# (sympy.Symbol('c5'),c5),
# (sympy.Symbol('c6'),c6),
# (sympy.Symbol('c7'),c7),
# (sympy.Symbol('c8'),c8),
# (sympy.Symbol('c9'),c9),
# (sympy.Symbol('c10'),c10),
# (sympy.Symbol('c11'),c11),
# (sympy.Symbol('c12'),c12)]))
#print("test7",psi_x.subs([
# (x,1.-delt*epsi),
# (y,kapp*epsi),
# (sympy.Symbol('c1'),c1),
# (sympy.Symbol('c2'),c2),
# (sympy.Symbol('c3'),c3),
# (sympy.Symbol('c4'),c4),
# (sympy.Symbol('c5'),c5),
# (sympy.Symbol('c6'),c6),
# (sympy.Symbol('c7'),c7),
# (sympy.Symbol('c8'),c8),
# (sympy.Symbol('c9'),c9),
# (sympy.Symbol('c10'),c10),
# (sympy.Symbol('c11'),c11),
# (sympy.Symbol('c12'),c12)]))
#print("test8",psi_x.subs([
# (x,xsep),
# (y,ysep),
# (sympy.Symbol('c1'),c1),
# (sympy.Symbol('c2'),c2),
# (sympy.Symbol('c3'),c3),
# (sympy.Symbol('c4'),c4),
# (sympy.Symbol('c5'),c5),
# (sympy.Symbol('c6'),c6),
# (sympy.Symbol('c7'),c7),
# (sympy.Symbol('c8'),c8),
# (sympy.Symbol('c9'),c9),
# (sympy.Symbol('c10'),c10),
# (sympy.Symbol('c11'),c11),
# (sympy.Symbol('c12'),c12)]))
#print("test9",psi_y.subs([
# (x,xsep),
# (y,ysep),
# (sympy.Symbol('c1'),c1),
# (sympy.Symbol('c2'),c2),
# (sympy.Symbol('c3'),c3),
# (sympy.Symbol('c4'),c4),
# (sympy.Symbol('c5'),c5),
# (sympy.Symbol('c6'),c6),
# (sympy.Symbol('c7'),c7),
# (sympy.Symbol('c8'),c8),
# (sympy.Symbol('c9'),c9),
# (sympy.Symbol('c10'),c10),
# (sympy.Symbol('c11'),c11),
# (sympy.Symbol('c12'),c12)]))
#print("test10",(psi_yy+N1*psi_x).subs([
# (x,1.+epsi),
# (y,0.),
# (sympy.Symbol('c1'),c1),
# (sympy.Symbol('c2'),c2),
# (sympy.Symbol('c3'),c3),
# (sympy.Symbol('c4'),c4),
# (sympy.Symbol('c5'),c5),
# (sympy.Symbol('c6'),c6),
# (sympy.Symbol('c7'),c7),
# (sympy.Symbol('c8'),c8),
# (sympy.Symbol('c9'),c9),
# (sympy.Symbol('c10'),c10),
# (sympy.Symbol('c11'),c11),
# (sympy.Symbol('c12'),c12)]))
#print("test11",(psi_yy+N2*psi_x).subs([
# (x,1.-epsi),
# (y,0.),
# (sympy.Symbol('c1'),c1),
# (sympy.Symbol('c2'),c2),
# (sympy.Symbol('c3'),c3),
# (sympy.Symbol('c4'),c4),
# (sympy.Symbol('c5'),c5),
# (sympy.Symbol('c6'),c6),
# (sympy.Symbol('c7'),c7),
# (sympy.Symbol('c8'),c8),
# (sympy.Symbol('c9'),c9),
# (sympy.Symbol('c10'),c10),
# (sympy.Symbol('c11'),c11),
# (sympy.Symbol('c12'),c12)]))
#print("test12",(psi_xx+N3*psi_y).subs([
# (x,1.-delt*epsi),
# (y,kapp*epsi),
# (sympy.Symbol('c1'),c1),
# (sympy.Symbol('c2'),c2),
# (sympy.Symbol('c3'),c3),
# (sympy.Symbol('c4'),c4),
# (sympy.Symbol('c5'),c5),
# (sympy.Symbol('c6'),c6),
# (sympy.Symbol('c7'),c7),
# (sympy.Symbol('c8'),c8),
# (sympy.Symbol('c9'),c9),
# (sympy.Symbol('c10'),c10),
# (sympy.Symbol('c11'),c11),
# (sympy.Symbol('c12'),c12)]))
def setPlotLimits(self,xmin,xmax,ymin,ymax):
"""
Set limits for plotting of flux surfaces, in x,y coordinates.
"""
self.xmin = xmin
self.xmax = xmax
self.ymin = ymin
self.ymax = ymax
def plotFluxSurfaces(self):
"""
Plot the flux surfaces from computed solutions.
"""
x1 = numpy.linspace(self.xmin,self.xmax,100)
y1 = numpy.linspace(self.ymin,self.ymax,100)
psiArray = numpy.zeros((100,100))
psi = self._psi_xy()
for i in range(100):
psiArray[:,i] = psi(x1[i],y1)
if self.Psiaxis is None:
psiAxis = self._getpsiAxisForPlotting()
else:
psiAxis = self.Psiaxis/self.Psi0
pyplot.contour(x1,y1,psiArray,numpy.linspace(-.1*psiAxis,psiAxis,15))
pyplot.show()
def Psi(self):
"""
Returns a function that evaluates the poloidal flux Psi at (R,Z)
"""
R,Z = sympy.symbols("R Z")
return sympy.lambdify((R,Z), self.Psi0*self._getpsi(R/self.R0,Z/self.R0,self.A,self.c1,self.c2,self.c3,self.c4,self.c5,self.c6,self.c7,self.c8,self.c9,self.c10,self.c11,self.c12),"numpy")
def dPsidR(self):
"""
Returns a function that evaluates the R-derivative of the poloidal flux dPsi/dR at (R,Z)
"""
return lambda R,Z: self.Psi0*self._dpsidx_xy()(R/self.R0,Z/self.R0)/self.R0
def dPsidZ(self):
"""
Returns a function that evaluates the Z-derivative of the poloidal flux dPsi/dZ at (R,Z)
"""
return lambda R,Z: self.Psi0*self._dpsidy_xy()(R/self.R0,Z/self.R0)/self.R0
def getAxis(self):
"""
Finds the maximum of Psi, which is the magnetic axis. Sets the values of Raxis, Zaxis and Psiaxis, and also returns them.
"""
#if (not self.Raxis) or (not self.Zaxis) or (not self.Psiaxis):
result = scipy.optimize.minimize(lambda posArray: self._psi_xy()(posArray[0],posArray[1]),[1.,0.])
x = result.x[0]
y = result.x[1]
psi = self._psi_xy()(x,y)
self.Raxis = x*self.R0
self.Zaxis = y*self.R0
self.Psiaxis = psi*self.Psi0
return self.Raxis, self.Zaxis, self.Psiaxis
def _getpsiAxisForPlotting(self):
"""
Finds the maximum of Psi, which is the magnetic axis. Sets the values of Raxis, Zaxis and Psiaxis, and also returns them.
"""
#if (not self.Raxis) or (not self.Zaxis) or (not self.Psiaxis):
result = scipy.optimize.minimize(lambda posArray: self._psi_xy()(posArray[0],posArray[1]),[1.,0.])
x = result.x[0]
y = result.x[1]
psi = self._psi_xy()(x,y)
return psi
def _psi_xy(self):
# normalised form
x,y = sympy.symbols("x y")
return sympy.lambdify((x,y), self._getpsi(x,y,self.A,self.c1,self.c2,self.c3,self.c4,self.c5,self.c6,self.c7,self.c8,self.c9,self.c10,self.c11,self.c12),"numpy")
def _dpsidx_xy(self):
x,y = sympy.symbols("x y")
psi = self._getpsi(x,y,self.A,self.c1,self.c2,self.c3,self.c4,self.c5,self.c6,self.c7,self.c8,self.c9,self.c10,self.c11,self.c12)
return sympy.lambdify((x,y),sympy.diff(psi,x),"numpy")
def _dpsidy_xy(self):
x,y = sympy.symbols("x y")
psi = self._getpsi(x,y,self.A,self.c1,self.c2,self.c3,self.c4,self.c5,self.c6,self.c7,self.c8,self.c9,self.c10,self.c11,self.c12)
return sympy.lambdify((x,y),sympy.diff(psi,y),"numpy")
def _psi_xy_symbolic(self,x,y):
return self._getpsi(x,y,self.A,self.c1,self.c2,self.c3,self.c4,self.c5,self.c6,self.c7,self.c8,self.c9,self.c10,self.c11,self.c12)
def _dpsidx_xy_symbolic(self,x,y):
return sympy.diff(self._psi_xy_symbolic(x,y),x)
def _dpsidy_xy_symbolic(self,x,y):
return sympy.diff(self._psi_xy_symbolic(x,y),y)
def _pOverPsi02_xy(self):
return lambda x,y: -1./self.mu0/self.R0**4*(1.-self.A)*self._psi_xy()(x,y)
def _BpOverPsi0_xy(self):
return lambda x,y: 1./self.R0*numpy.sqrt(self._dpsidx_xy()(x,y)**2+self._dpsidy_xy()(x,y)**2) / x / self.R0
def _findSeparatrix(self,theta):
# Find minor radius of psi=0 in normalised (x-y) units
psi = self._psi_xy()
psiAxis = psi(1.,0.) # Not really the axis, but should be close enough
sign = psiAxis/numpy.abs(psiAxis)
psiAtTheta = lambda a: sign*psi(1.+a*numpy.cos(theta),a*numpy.sin(theta)) # sign so that this function always decreases away from the axis
# Need to find an endpoint with positive psi
rguess = ( 2.*self.epsi*numpy.cos(theta)**2 # catch inner and outer equatorial points which are at 1+epsi for theta=0 and 1-epsi for theta=2pi
+ 2.*self.kapp*self.epsi*numpy.sin(theta)**2 # catch high point and low point which are at kapp*epsi and -kapp*epsi
)
if rguess*numpy.cos(theta)<-1.:
# Avoid having x=0.
rguess = -.999/numpy.cos(theta)
if psi(1.+rguess*numpy.cos(theta),rguess*numpy.sin(theta))*psiAxis<0.: # psi changes sign
endpoint = rguess
else:
result = scipy.optimize.minimize_scalar(psiAtTheta, bounds=(0.,rguess),method='bounded')
a_max = result.x
psi_max = psi(1.+a_max*numpy.cos(theta),a_max*numpy.sin(theta))
if psi_max*psiAxis<0.:
endpoint = a_max
else:
# Very close to the X-point, just use a_max as the minor radius of psi=0 surface
return a_max
return scipy.optimize.brentq(psiAtTheta,0.,endpoint)
def _betatOverPsi02(self):
pOverPsi02 = self._pOverPsi02_xy()
pOverPsi02_total = 0.
volume = 0.
Ntheta = self.integrationGridSize
dtheta = 2.*numpy.pi/float(Ntheta)
for theta in numpy.linspace(0.,2.*numpy.pi,Ntheta,endpoint=False):
a = self._findSeparatrix(theta)
integral,error = scipy.integrate.quad(
lambda r: pOverPsi02(1.+r*numpy.cos(theta),r*numpy.sin(theta)) * r * (1.+r*numpy.cos(theta))*2.*numpy.pi
,0.,a)
pOverPsi02_total += integral*dtheta
#integral,error = scipy.integrate.quad(lambda r: r * (1.+r*numpy.cos(theta))*2.*numpy.pi,0.,a)
integral = (a**2/2.+a**3/3.*numpy.cos(theta))*2.*numpy.pi
volume += integral*dtheta
pOverPsi02_average = pOverPsi02_total/volume
return 2.*self.mu0*pOverPsi02_average/self.B0**2
def betat(self):
"""
Returns the toroidal beta
"""
return self.Psi0**2*self._betatOverPsi02()
def calculatePsi0FromBetat(self,betat):
"""
Calculates Psi0 given an input toroidal beta
"""
self.Psi0 = numpy.sqrt(betat/self._betatOverPsi02())
self.getAxis() # Now that Psi0 is known, can compute Psiaxis, so call getAxis now
return self.Psi0
def calculateCp(self):
"""
Calculates the normalised plasma circumference Cp
"""
Ntheta = self.integrationGridSize
dtheta = 2.*numpy.pi/float(Ntheta)
self.Cp = 0.
aplus = self._findSeparatrix(-dtheta/2.)
xplus = 1.+aplus*numpy.cos(-dtheta/2.)
yplus = aplus*numpy.sin(-dtheta/2.)
for theta in numpy.linspace(0.,2.*numpy.pi,Ntheta,endpoint=False):
aminus = aplus
xminus = xplus
yminus = yplus
aplus = self._findSeparatrix(theta+dtheta/2.)
xplus = 1.+aplus*numpy.cos(theta+dtheta/2.)
yplus = aplus*numpy.sin(theta+dtheta/2.)
self.Cp += numpy.sqrt((xplus-xminus)**2+(yplus-yminus)**2)
return self.Cp
def _integralBpOverPsi0AroundSeparatrix(self):
# loop integral of Bp over plasma surface
# Integral with unnormalised lengths (eventually): i.e. includes factor of R0
psi = self._psi_xy()
psi_x = self._dpsidx_xy()
psi_y = self._dpsidy_xy()
Ntheta = self.integrationGridSize
dtheta = 2.*numpy.pi/float(Ntheta)
integralBpOverPsi0 = 0.
aplus = self._findSeparatrix(-dtheta/2.)
xplus = 1.+aplus*numpy.cos(-dtheta/2.)
yplus = aplus*numpy.sin(-dtheta/2.)
for theta in numpy.linspace(0.,2.*numpy.pi,Ntheta,endpoint=False):
aminus = aplus
xminus = xplus
yminus = yplus
aplus = self._findSeparatrix(theta+dtheta/2.)
xplus = 1.+aplus*numpy.cos(theta+dtheta/2.)
yplus = aplus*numpy.sin(theta+dtheta/2.)
a = self._findSeparatrix(theta)
x = 1.+a*numpy.cos(theta)
y = a*numpy.sin(theta)
this_psi_x = psi_x(x,y)
this_psi_y = psi_y(x,y)
dlOverR0 = numpy.sqrt((xplus-xminus)**2+(yplus-yminus)**2)
integralBpOverPsi0 += numpy.sqrt(this_psi_x**2+this_psi_y**2)/x*dlOverR0 # sqrt(psi_x**2+psi_y**2)=R0**2*x*(Bp/Psi0)
# Divide out left-over factor of R0
integralBpOverPsi0 /= self.R0
return integralBpOverPsi0
def calculatePsi0FromCurrent(self,I):
"""
Calculates Psi0 given a value for the plasma current
"""
# Bpbar = mu0*I/R0/Cp => integral(Bp) = mu0*I
self.Psi0 = self.mu0*I/self._integralBpOverPsi0AroundSeparatrix()
self.getAxis() # Now that Psi0 is known, can compute Psiaxis, so call getAxis now
return self.Psi0
def _testfuncBetatAndCurrent(self,A,betat,I):
print("trying ",A)
# Re-initialise with this value of A
self.init(self.epsi,self.kapp,self.delt,self.xsep,self.ysep,A,self.R0,self.B0,self.Psi0)
# Using the current as the test variable (solving for Psi0 with betat) seems to converge better??
print("Psi0",self.calculatePsi0FromCurrent(I))
currentbetat = self.betat()
print("betat",currentbetat)
print("testval",currentbetat-betat)
print("")
#return self.betat()-betat
return currentbetat - betat
#print("Psi0",self.calculatePsi0FromBetat(betat))
#thisCurrent = self.current()
#print("current",thisCurrent)
#print("testval",thisCurrent - I)
#return thisCurrent - I
def _calculateAAndPsi0(self,testfunc,arg1,arg2):
# Use given value of A as an initial guess, and find a root that gives the desired optimization quantities
# Start with low resolution to get close
origN = self.integrationGridSize
self.integrationGridSize = 50
Amax = .9999
searching = True
testlower = -1.
testupper = Amax # Don't need to do this if betat is zero, since then A is 1. exactly, but checking A=1. when calculating Psi0 from betat gives a divide-by-zero error, so start just below 1.
while searching:
try:
self.A = scipy.optimize.brentq(testfunc,testupper,testlower,args=(arg1,arg2))
searching = False
except ValueError as e:
# Presumably upper and lower limits have the same sign of the test value, so use a new range
testupper = testlower
testlower *= 2.
if testlower>=testupper:
raise ValueError("Failed to find good limits")
print("changing range to search to ("+str(testlower)+","+str(testupper)+")")
# Now refine with high resolution, assuming that we are within testInterval of the accurate solution already
testInterval = 0.04
self.integrationGridSize = origN
firstA = self.A
while True:
try:
testlower = firstA-testInterval
testupper = firstA+testInterval
if testupper>Amax:
testupper = Amax
self.A = scipy.optimize.brentq(testfunc,testlower,testupper,args=(arg1,arg2))
break
except ValueError as error:
if error.message == "f(a) and f(b) must have different signs":
# Presumably interval was too small, so try increasing it
if testInterval > 2.:
# Interval is bigger than the original one, something must have gone wrong
error.message = "Unexpected failure to find interval around zero of A."
raise error
else:
testInterval *= 2.
print("testInterval",testInterval)
print("found A =",self.A,"and Psi0 =",self.Psi0,"for arg1 =",arg1,"and arg2 =",arg2)
def calculateAAndPsi0FromBetatAndCurrent(self,betat,I):
"""
Iteratively calculate the integration constant A and the value of Psi0 from toroidal beta, betat, and plasma current, I.
"""
self._calculateAAndPsi0(self._testfuncBetatAndCurrent,betat,I)
def q(self,psiVal):
"""
Calculate safety factor, q for some psi-surface
"""
# Integral with unnormalised lengths (eventually): i.e. includes factor of R0
Ntheta = self.integrationGridSize
dtheta = 2.*numpy.pi/float(Ntheta)
thetaGrid = numpy.linspace(0.,2.*numpy.pi,Ntheta,endpoint=False)
R,Z,r = self.getFluxSurfaceGrid([psiVal],thetaGrid[:])
Bt = self.Bt()(R,Z)
dthetadR = -numpy.sin(thetaGrid)/r
dthetadZ = numpy.cos(thetaGrid)/r
JTimesR = (self.dPsidR()(R,Z)*dthetadZ-self.dPsidZ()(R,Z)*dthetadR) # J=B.Grad(theta)=Bp.Grad(theta)
q_integral = (Bt/JTimesR).sum()*dtheta
# Divide by integral(dtheta)
q_integral /= 2.*numpy.pi
return q_integral
def _testfuncBetatAndq(self,A,betat,q,psiVal):
print("trying ",A)
# Re-initialise with this value of A
self.init(self.epsi,self.kapp,self.delt,self.xsep,self.ysep,A,self.R0,self.B0,self.Psi0)
# Using the current as the test variable (solving for Psi0 with betat) seems to converge better??
print("Psi0",self.calculatePsi0FromBetat(betat))
currentq = self.q(psiVal)
print("q",currentq)
print("testval",currentq-q)
print("")
return currentq-q
def calculateAAndPsi0FromBetatAndq(self,betat,q,psiVal):
"""
Iteratively calculate the integration constant A and the value of Psi0 from toroidal beta, betat, and the safety factor, q on a specified flux surface with psi of psiVal.
"""
self._calculateAAndPsi0(lambda A,betat,q: self._testfuncBetatAndq(A,betat,q,psiVal),betat,q)
def current(self):
"""
Calculate the plasma current from the equilibrium.
"""
return self.Psi0*self._integralBpOverPsi0AroundSeparatrix()/self.mu0
def p(self):
"""
Returns a function that evaluates the pressure at (R,Z)
"""
return lambda R,Z: -self.Psi0**2/self.mu0/self.R0**4*(1-self.A)*self._psi_xy()(R/self.R0,Z/self.R0)
def pressurePsiN(self, psinorm):
"""
Returns the pressure at normalised psi (0 = magnetic axis, 1 = plasma edge)
"""
psi = 1 - psinorm # psi in Cerfon-Freidberg 2010 goes from 1 on axis to 0 at edge
return self.Psi0**2 * (1. - self.A)*psi / (self.mu0*self.R0**4)
def fpolPsiN(self, psinorm):
"""
Returns the poloidal current function f = R*Bt at given normalised psi
(0 = magnetic axis, 1 = plasma edge)
"""
psi = 1 - psinorm # psi in Cerfon-Freidberg 2010 goes from 1 on axis to 0 at edge
return self.R0*numpy.sqrt(self.B0**2 - 2.*self.Psi0**2*self.A*psi/self.R0**4)
def Bt(self):
"""
Returns a function that evaluates the toroidal magnetic field at (R,Z)
"""
return lambda R,Z: numpy.sqrt(self.R0**2/R**2*(self.B0**2-2.*self.Psi0**2/self.R0**4*self.A*self._psi_xy()(R/self.R0,Z/self.R0)))
def Bt_symbolic(self,R,Z):
"""
Returns a sympy expression for the toroidal magnetic field in terms of x and y
"""
x,y = sympy.symbols("x y")
return sympy.sqrt(self.R0**2/R**2*(self.B0**2-2.*self.Psi0**2/self.R0**4*self.A*self._psi_xy_symbolic(x,y).subs([(x,R/self.R0),(y,Z/self.R0)])))
def dBtdR(self):
"""
Returns a function that evaluates the R derivative of the toroidal magnetic field at (R,Z)
"""
R,Z = sympy.symbols("R Z")
return sympy.lambdify((R,Z),sympy.diff(self.Bt_symbolic(R,Z),R),"numpy")
def dBtdZ(self):
"""
Returns a function that evaluates the Z derivative of the toroidal magnetic field at (R,Z)
"""
R,Z = sympy.symbols("R Z")
return sympy.lambdify((R,Z),sympy.diff(self.Bt_symbolic(R,Z),Z),"numpy")
def Bp(self):
"""
Returns a function that evaluates the poloidal magnetic field at (R,Z)
"""
return lambda R,Z: self.Psi0/self.R0*numpy.sqrt(self._dpsidx_xy()(R/self.R0,Z/self.R0)**2+self._dpsidy_xy()(R/self.R0,Z/self.R0)**2) / R
def Bp_symbolic(self,R,Z):
"""
Returns a sympy expression for the poloidal magnetic field in terms of x and y
"""
x,y = sympy.symbols("x y")
return self.Psi0/self.R0*sympy.sqrt(self._dpsidx_xy_symbolic(x,y).subs([(x,R/self.R0),(y,Z/self.R0)])**2+self._dpsidy_xy_symbolic(x,y).subs([(x,R/self.R0),(y,Z/self.R0)])**2) / R
def dBpdR(self):
"""
Returns a function that evaluates the R derivative of the poloidal magnetic field at (R,Z)
"""
R,Z = sympy.symbols("R Z")
return sympy.lambdify((R,Z),sympy.diff(self.Bp_symbolic(R,Z),R),"numpy")
def dBpdZ(self):
"""
Returns a function that evaluates the Z derivative of the poloidal magnetic field at (R,Z)
"""
R,Z = sympy.symbols("R Z")
return sympy.lambdify((R,Z),sympy.diff(self.Bp_symbolic(R,Z),Z),"numpy")
def BR(self):
"""
Returns a function that evaluates the R component the magnetic field at (R,Z)
"""
return lambda R,Z: self.Psi0/self.R0*self._dpsidx_xy()(R/self.R0,Z/self.R0) / R
def BZ(self):
"""
Returns a function that evaluates the Z component the magnetic field at (R,Z)
"""
return lambda R,Z: self.Psi0/self.R0*self._dpsidy_xy()(R/self.R0,Z/self.R0) / R
def getMinorRadiusGrid(self,theta,psiNGrid,rguess=None):
"""
Find the minor radius for an array of psiN at some angle theta
"""
if psiNGrid.ndim > 1:
raise ValueError("psiNGrid has more than one dimension.")
if psiNGrid.max() >= 1.:
raise ValueError("psiNGrid contains value(s) greater than or equal to 1")
if psiNGrid.min() <= 0.:
raise ValueError("psiNGrid contains value(s) less than or equal to 0")
if psiNGrid.ndim == 0:
# enumerate cannot handle 0d array
psiNGrid = psiNGrid[numpy.newaxis]
PsiGrid = (1.-psiNGrid)*self.Psiaxis
# Function to evalute Psi for a given minor radius
Psi = self.Psi()
sign = self.Psiaxis/numpy.abs(self.Psiaxis)
PsiAtTheta = lambda r: Psi(self.Raxis+r*numpy.cos(theta),self.Zaxis+r*numpy.sin(theta)) # sign so that this function always decreases away from the axis
# Find an endpoint where Psi has the opposite sign to Psiaxis
##if PsiAtTheta(self.epsi*self.R0)*self.Psiaxis<0.:
## endpoint = self.epsi*self.R0
##else:
## result = scipy.optimize.minimize_scalar(lambda r: sign*PsiAtTheta(r), bounds=(0.,2.*self.R0),method='bounded')
## endpoint = result.x
###endpoint = self.R0*self._findSeparatrix(theta) # Assuming we are only interested in closed field lines, this avoids duplicating heuristics in _findSeparatrix
# Need to find an endpoint with positive psi
if not rguess:
rguess = ( 2.*(self.R0*self.epsi-(self.Raxis-self.R0)*numpy.cos(theta))*numpy.cos(theta)**2 # catch inner and outer equatorial points which are at 1+epsi for theta=0 and 1-epsi for theta=2pi
+ 2.*(self.R0*self.kapp*self.epsi-self.Zaxis*numpy.sin(theta))*numpy.sin(theta)**2 # catch high point and low point which are at kapp*epsi and -kapp*epsi
)
if rguess*numpy.cos(theta)<-self.Raxis:
#print("option A")
# Avoid having R=0.
rguess = -.999*self.Raxis/numpy.cos(theta)
if PsiAtTheta(rguess)*self.Psiaxis<0.: # psi changes sign
#print("option B")
endpoint = rguess
else:
#print("option C, rguess:",rguess)
result = scipy.optimize.minimize_scalar(lambda r: PsiAtTheta(r)/self.Psiaxis, bounds=(0.,rguess),method='bounded')
r_max = result.x
Psi_max = PsiAtTheta(r_max)
endpoint = r_max
#print("")
#aval = self._findSeparatrix(theta)
#psia = self._psi_xy()(1.+aval*numpy.cos(theta),aval*numpy.sin(theta))
#print("aval",aval)
#print("psia",psia)
#print("Psia",Psi(self.R0*(1.+aval*numpy.cos(theta)),self.R0*aval*numpy.sin(theta)))
#print("")
#print("endpoint",endpoint)
#print("theta",theta)
#print("Psi at 0:",PsiAtTheta(0.),PsiAtTheta(0.)-PsiGrid[0])
#print("Psi at endpoint:",PsiAtTheta(endpoint),PsiAtTheta(endpoint)-PsiGrid[0])
##print("Psi near endpoint",PsiAtTheta(endpoint-.1),PsiAtTheta(endpoint+.1))
#print("PsiGrid",PsiGrid)
#print("PsiNGrid",psiNGrid)
#print("Psiaxis",self.Psiaxis)
#print("RAxis",self.Raxis)
#print("ZAxis",self.Zaxis)
rGrid = numpy.zeros(PsiGrid.size)
for i,PsiVal in enumerate(PsiGrid):
rGrid[i] = scipy.optimize.brentq(lambda r: PsiAtTheta(r)-PsiVal,0.,endpoint)
return rGrid
def getFluxSurfaceGrid(self,psiNGrid,thetaGrid):
"""
Take 1d arrays of normalised flux, psiNGrid, and poloidal angle (centred on the magnetic axis), thetaGrid,
and return a 2d arrays giving major radius, R[ipsi,itheta], and height, Z[ipsi,itheta], the Cartesian
coordinates in the poloidal plane, and also the minor radius, r[ipsi,itheta], of the logically rectangular
psiNGrid*thetaGrid grid.
"""
psiNGrid = numpy.array(psiNGrid)
thetaGrid = numpy.array(thetaGrid)
if psiNGrid.ndim > 1:
raise ValueError("psiN has more than one dimension.")
if thetaGrid.ndim > 1:
raise ValueError("theta has more than one dimension.")
if psiNGrid.max() >= 1.:
raise ValueError("psiNGrid contains value(s) greater than or equal to 1")
if psiNGrid.min() <= 0.:
raise ValueError("psiNGrid contains value(s) less than or equal to 0")
if thetaGrid.ndim == 0:
# enumerate cannot handle 0d array
thetaGrid = thetaGrid[numpy.newaxis]
Npsi = psiNGrid.size
Ntheta = thetaGrid.size
R = numpy.zeros([Npsi,Ntheta])
Z = numpy.zeros([Npsi,Ntheta])
r = numpy.zeros([Npsi,Ntheta])
rguess = None
for itheta,t in enumerate(thetaGrid):
r[:,itheta] = self.getMinorRadiusGrid(t,psiNGrid,rguess)
rguess = ( 1.1 + 2.*(thetaGrid[(itheta+1)%Ntheta]-t) )*r[-1,itheta] # minor radius should not change too much between adjacent theta-grid points.
R = self.Raxis + r*numpy.cos(thetaGrid[numpy.newaxis,:])
Z = self.Zaxis + r*numpy.sin(thetaGrid[numpy.newaxis,:])
return R,Z,r
@staticmethod
def _getpsi(x,y,A,c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12): # get parts of psi
p = ( 1./8.*x**4 + A*(x**2.*sympy.log(x)/2.-1./8.*x**4) + c1*1. + c2*x**2 + c3*(y**2 - x**2*sympy.log(x))
+ c4*(x**4 - 4.*x**2*y**2) + c5*(2.*y**4 - 9.*y**2*x**2
+ 3.*x**4*sympy.log(x) - 12.*x**2*y**2*sympy.log(x)) + c6*(x**6
- 12.*x**4*y**2 + 8.*x**2*y**4) + c7*(8.*y**6 - 140.*y**4*x**2
+ 75.*y**2*x**4 - 15*x**6*sympy.log(x) + 180.*x**4*y**2*sympy.log(x)
- 120.*x**2*y**4*sympy.log(x)) + c8*(y) + c9*(y*x**2)
+ c10*(y**3 - 3.*y*x**2*sympy.log(x)) + c11*(3.*y*x**4 - 4.*y**3*x**2)
+ c12*(8.*y**5 - 45.*y*x**4 - 80.*y**3*x**2*sympy.log(x)
+ 60.*y*x**4*sympy.log(x))
)
return p