def subTracers(q, vv, p, tracers0, iproc, hMin = 2e-3, hMax = 2e4, lMax = 500, tol = 1e-2,
interpolation = 'weighted', integration = 'simple', intQ = ['']):
tracers = tracers0
mapping = np.zeros((tracers.shape[0], tracers.shape[1], 3))
for ix in range(tracers.shape[0]):
for iy in range(tracers.shape[1]):
xx = tracers[ix, iy, 2:5].copy()
s = pc.stream(vv, p, interpolation = interpolation, integration = integration, hMin = hMin, hMax = hMax, lMax = lMax, tol = tol, xx = xx)
tracers[ix, iy, 2:5] = s.tracers[s.sl-1]
tracers[ix, iy, 5] = s.l
if (any(intQ == 'curlyA')):
for l in range(s.sl-1):
aaInt = pc.vecInt((s.tracers[l+1] + s.tracers[l])/2, aa, p, interpolation)
tracers[ix, iy, 6] += np.dot(aaInt, (s.tracers[l+1] - s.tracers[l]))
# create the color mapping
if (tracers[ix, iy, 4] > grid.z[-2]):
if (tracers[ix, iy, 0] - tracers[ix, iy, 2]) > 0:
if (tracers[ix, iy, 1] - tracers[ix, iy, 3]) > 0:
mapping[ix, iy, :] = [0,1,0]
else:
mapping[ix, iy, :] = [1,1,0]
else:
if (tracers[ix, iy, 1] - tracers[ix, iy, 3]) > 0:
mapping[ix, iy, :] = [0,0,1]
else:
mapping[ix, iy, :] = [1,0,0]
else:
mapping[ix, iy, :] = [1,1,1]
q.put((tracers, mapping, iproc))
def subTracers(q,
vv,
p,
tracers0,
iproc,
hMin=2e-3,
hMax=2e4,
lMax=500,
tol=1e-2,
interpolation='weighted',
integration='simple',
intQ=['']):
tracers = tracers0
mapping = np.zeros((tracers.shape[0], tracers.shape[1], 3))
for ix in range(tracers.shape[0]):
for iy in range(tracers.shape[1]):
xx = tracers[ix, iy, 2:5].copy()
s = pc.stream(vv,
p,
interpolation=interpolation,
integration=integration,
hMin=hMin,
hMax=hMax,
lMax=lMax,
tol=tol,
xx=xx)
tracers[ix, iy, 2:5] = s.tracers[s.sl - 1]
tracers[ix, iy, 5] = s.l
if (any(intQ == 'curlyA')):
for l in range(s.sl - 1):
aaInt = pc.vecInt(
(s.tracers[l + 1] + s.tracers[l]) / 2, aa, p,
interpolation)
tracers[ix, iy,
6] += np.dot(aaInt,
(s.tracers[l + 1] - s.tracers[l]))
# create the color mapping
if (tracers[ix, iy, 4] > grid.z[-2]):
if (tracers[ix, iy, 0] - tracers[ix, iy, 2]) > 0:
if (tracers[ix, iy, 1] - tracers[ix, iy, 3]) > 0:
mapping[ix, iy, :] = [0, 1, 0]
else:
mapping[ix, iy, :] = [1, 1, 0]
else:
if (tracers[ix, iy, 1] - tracers[ix, iy, 3]) > 0:
mapping[ix, iy, :] = [0, 0, 1]
else:
mapping[ix, iy, :] = [1, 0, 0]
else:
mapping[ix, iy, :] = [1, 1, 1]
q.put((tracers, mapping, iproc))
def edge(vv,
p,
sx,
sy,
diff1,
diff2,
phiMin,
rec,
hMin=hMin,
hMax=hMax,
lMax=lMax,
tol=tol,
interpolation=interpolation,
integration=integration):
dtot = m.atan2(diff1[0] * diff2[1] - diff2[0] * diff1[1],
diff1[0] * diff2[0] + diff1[1] * diff2[1])
if ((abs(dtot) > phiMin) and (rec < 4)):
xm = 0.5 * (sx[0] + sx[1])
ym = 0.5 * (sy[0] + sy[1])
# trace intermediate field line
s = pc.stream(vv,
p,
hMin=hMin,
hMax=hMax,
lMax=lMax,
tol=tol,
interpolation=interpolation,
integration=integration,
xx=np.array([xm, ym, p.Oz]))
tracer = np.concatenate(
(s.tracers[0,
0:2], s.tracers[s.sl - 1, :], np.reshape(s.l, (1))))
# discard any streamline which does not converge or hits the boundary
if ((tracer[5] >= lMax) or (tracer[4] < p.Oz + p.Lz - p.dz)):
dtot = 0.
else:
diffm = np.array(
[tracer[2] - tracer[0], tracer[3] - tracer[1]])
if (sum(diffm**2) != 0):
diffm = diffm / np.sqrt(sum(diffm**2))
dtot = edge(vv, p, [sx[0], xm], [sy[0], ym], diff1, diffm, phiMin, rec+1,
hMin = hMin, hMax = hMax, lMax = lMax, tol = tol, interpolation = interpolation, integration = integration)+ \
edge(vv, p, [xm, sx[1]], [ym, sy[1]], diffm, diff2, phiMin, rec+1,
hMin = hMin, hMax = hMax, lMax = lMax, tol = tol, interpolation = interpolation, integration = integration)
return dtot
def edge(vv, p, sx, sy, diff1, diff2, phiMin, rec, hMin = hMin, hMax = hMax,
lMax = lMax, tol = tol, interpolation = interpolation, integration = integration):
dtot = m.atan2(diff1[0]*diff2[1] - diff2[0]*diff1[1], diff1[0]*diff2[0] + diff1[1]*diff2[1])
if ((abs(dtot) > phiMin) and (rec < 4)):
xm = 0.5*(sx[0]+sx[1])
ym = 0.5*(sy[0]+sy[1])
# trace intermediate field line
s = pc.stream(vv, p, hMin = hMin, hMax = hMax, lMax = lMax, tol = tol,
interpolation = interpolation, integration = integration, xx = np.array([xm, ym, p.Oz]))
tracer = np.concatenate((s.tracers[0,0:2], s.tracers[s.sl-1,:], np.reshape(s.l,(1))))
# discard any streamline which does not converge or hits the boundary
if ((tracer[5] >= lMax) or (tracer[4] < p.Oz+p.Lz-p.dz)):
dtot = 0.
else:
diffm = np.array([tracer[2] - tracer[0], tracer[3] - tracer[1]])
if (sum(diffm**2) != 0):
diffm = diffm / np.sqrt(sum(diffm**2))
dtot = edge(vv, p, [sx[0], xm], [sy[0], ym], diff1, diffm, phiMin, rec+1,
hMin = hMin, hMax = hMax, lMax = lMax, tol = tol, interpolation = interpolation, integration = integration)+ \
edge(vv, p, [xm, sx[1]], [ym, sy[1]], diffm, diff2, phiMin, rec+1,
hMin = hMin, hMax = hMax, lMax = lMax, tol = tol, interpolation = interpolation, integration = integration)
return dtot
def subFixed(queue, ix0, iy0, vv, p, tracers, iproc, hMin = 2e-3, hMax = 2e4, lMax = 500,
tol = 1e-2, interpolation = 'weighted', integration = 'simple'):
diff = np.zeros((4,2))
phiMin = np.pi/8.
x = []
y = []
q = []
fidx = 0
for ix in ix0:
for iy in iy0:
# compute Poincare index around this cell (!= 0 for potential fixed point)
diff[0,:] = tracers[iy, ix, 0, 2:4] - tracers[iy, ix, 0, 0:2]
diff[1,:] = tracers[iy, ix+1, 0, 2:4] - tracers[iy, ix+1, 0, 0:2]
diff[2,:] = tracers[iy+1, ix+1, 0, 2:4] - tracers[iy+1, ix+1, 0, 0:2]
diff[3,:] = tracers[iy+1, ix, 0, 2:4] - tracers[iy+1, ix, 0, 0:2]
if (sum(np.sum(diff**2, axis = 1) != 0) == True):
diff = np.swapaxes(np.swapaxes(diff, 0, 1) / np.sqrt(np.sum(diff**2, axis = 1)), 0, 1)
poincare = pIndex(vv, p, tracers[iy, ix:ix+2, 0, 0], tracers[iy:iy+2, ix, 0, 1], diff, phiMin,
hMin = hMin, hMax = hMax, lMax = lMax, tol = tol, interpolation = interpolation, integration = integration)
if (abs(poincare) > 5): # use 5 instead of 2pi to account for rounding errors
# subsample to get starting point for iteration
nt = 4
xmin = tracers[iy, ix, 0, 0]
ymin = tracers[iy, ix, 0, 1]
xmax = tracers[iy, ix+1, 0, 0]
ymax = tracers[iy+1, ix, 0, 1]
xx = np.zeros((nt**2,3))
tracersSub = np.zeros((nt**2,5))
i1 = 0
for j1 in range(nt):
for k1 in range(nt):
xx[i1,0] = xmin + j1/(nt-1.)*(xmax - xmin)
xx[i1,1] = ymin + k1/(nt-1.)*(ymax - ymin)
xx[i1,2] = p.Oz
i1 += 1
for it1 in range(nt**2):
s = pc.stream(vv, p, hMin = hMin, hMax = hMax, lMax = lMax, tol = tol,
interpolation = interpolation, integration = integration, xx = xx[it1,:])
tracersSub[it1,0:2] = xx[it1,0:2]
tracersSub[it1,2:] = s.tracers[s.sl-1,:]
min2 = 1e6
minx = xmin
miny = ymin
i1 = 0
for j1 in range(nt):
for k1 in range(nt):
diff2 = (tracersSub[i1, 2] - tracersSub[i1, 0])**2 + (tracersSub[i1, 3] - tracersSub[i1, 1])**2
if (diff2 < min2):
min2 = diff2
minx = xmin + j1/(nt-1.)*(xmax - xmin)
miny = ymin + k1/(nt-1.)*(ymax - ymin)
it1 += 1
# get fixed point from this starting position using Newton's method
#TODO:
dl = np.min(var.dx, var.dy)/100. # step-size for calculating the Jacobian by finite differences
it = 0
# tracers used to find the fixed point
tracersNull = np.zeros((5,4))
point = np.array([minx, miny])
while True:
# trace field lines at original point and for Jacobian:
# (second order seems to be enough)
xx = np.zeros((5,3))
xx[0,:] = np.array([point[0], point[1], p.Oz])
xx[1,:] = np.array([point[0]-dl, point[1], p.Oz])
xx[2,:] = np.array([point[0]+dl, point[1], p.Oz])
xx[3,:] = np.array([point[0], point[1]-dl, p.Oz])
xx[4,:] = np.array([point[0], point[1]+dl, p.Oz])
for it1 in range(5):
s = pc.stream(vv, p, hMin = hMin, hMax = hMax, lMax = lMax, tol = tol,
interpolation = interpolation, integration = integration, xx = xx[it1,:])
tracersNull[it1,:2] = xx[it1,:2]
tracersNull[it1,2:] = s.tracers[s.sl-1,0:2]
# check function convergence
ff = np.zeros(2)
ff[0] = tracersNull[0,2] - tracersNull[0,0]
ff[1] = tracersNull[0,3] - tracersNull[0,1]
#TODO:
if (sum(abs(ff)) <= 1e-4):
fixedPoint = np.array([point[0], point[1]])
break
# compute the Jacobian
fjac = np.zeros((2,2))
fjac[0,0] = ((tracersNull[2,2] - tracersNull[2,0]) - (tracersNull[1,2] - tracersNull[1,0]))/2./dl
fjac[0,1] = ((tracersNull[4,2] - tracersNull[4,0]) - (tracersNull[3,2] - tracersNull[3,0]))/2./dl
fjac[1,0] = ((tracersNull[2,3] - tracersNull[2,1]) - (tracersNull[1,3] - tracersNull[1,1]))/2./dl
fjac[1,1] = ((tracersNull[4,3] - tracersNull[4,1]) - (tracersNull[3,3] - tracersNull[3,1]))/2./dl
# invert the Jacobian
fjin = np.zeros((2,2))
det = fjac[0,0]*fjac[1,1] - fjac[0,1]*fjac[1,0]
#TODO:
if (abs(det) < dl):
fixedPoint = point
break
fjin[0,0] = fjac[1,1]
fjin[1,1] = fjac[0,0]
fjin[0,1] = -fjac[0,1]
fjin[1,0] = -fjac[1,0]
fjin = fjin/det
dpoint = np.zeros(2)
dpoint[0] = -fjin[0,0]*ff[0] - fjin[0,1]*ff[1]
dpoint[1] = -fjin[1,0]*ff[0] - fjin[1,1]*ff[1]
point += dpoint
# check root convergence
#TODO:
if (sum(abs(dpoint)) < 1e-4):
fixedPoint = point
break
if (it > 20):
fixedPoint = point
print("warning: Newton did not converged")
break
it += 1
# check if fixed point lies inside the cell
if ((fixedPoint[0] < tracers[iy, ix, 0, 0]) or (fixedPoint[0] > tracers[iy, ix+1, 0, 0]) or
(fixedPoint[1] < tracers[iy, ix, 0, 1]) or (fixedPoint[1] > tracers[iy+1, ix, 0, 1])):
print("warning: fixed point lies outside the cell")
else:
x.append(fixedPoint[0])
y.append(fixedPoint[1])
#q.append()
fidx += 1
queue.put((x, y, q, fidx, iproc))
def subFixed(queue,
ix0,
iy0,
vv,
p,
tracers,
iproc,
hMin=2e-3,
hMax=2e4,
lMax=500,
tol=1e-2,
interpolation='weighted',
integration='simple'):
diff = np.zeros((4, 2))
phiMin = np.pi / 8.
x = []
y = []
q = []
fidx = 0
for ix in ix0:
for iy in iy0:
# compute Poincare index around this cell (!= 0 for potential fixed point)
diff[0, :] = tracers[iy, ix, 0, 2:4] - tracers[iy, ix, 0, 0:2]
diff[1, :] = tracers[iy, ix + 1, 0, 2:4] - tracers[iy, ix + 1,
0, 0:2]
diff[2, :] = tracers[iy + 1, ix + 1, 0,
2:4] - tracers[iy + 1, ix + 1, 0, 0:2]
diff[3, :] = tracers[iy + 1, ix, 0, 2:4] - tracers[iy + 1, ix,
0, 0:2]
if (sum(np.sum(diff**2, axis=1) != 0) == True):
diff = np.swapaxes(
np.swapaxes(diff, 0, 1) /
np.sqrt(np.sum(diff**2, axis=1)), 0, 1)
poincare = pIndex(vv,
p,
tracers[iy, ix:ix + 2, 0, 0],
tracers[iy:iy + 2, ix, 0, 1],
diff,
phiMin,
hMin=hMin,
hMax=hMax,
lMax=lMax,
tol=tol,
interpolation=interpolation,
integration=integration)
if (abs(poincare) > 5
): # use 5 instead of 2pi to account for rounding errors
# subsample to get starting point for iteration
nt = 4
xmin = tracers[iy, ix, 0, 0]
ymin = tracers[iy, ix, 0, 1]
xmax = tracers[iy, ix + 1, 0, 0]
ymax = tracers[iy + 1, ix, 0, 1]
xx = np.zeros((nt**2, 3))
tracersSub = np.zeros((nt**2, 5))
i1 = 0
for j1 in range(nt):
for k1 in range(nt):
xx[i1, 0] = xmin + j1 / (nt - 1.) * (xmax - xmin)
xx[i1, 1] = ymin + k1 / (nt - 1.) * (ymax - ymin)
xx[i1, 2] = p.Oz
i1 += 1
for it1 in range(nt**2):
s = pc.stream(vv,
p,
hMin=hMin,
hMax=hMax,
lMax=lMax,
tol=tol,
interpolation=interpolation,
integration=integration,
xx=xx[it1, :])
tracersSub[it1, 0:2] = xx[it1, 0:2]
tracersSub[it1, 2:] = s.tracers[s.sl - 1, :]
min2 = 1e6
minx = xmin
miny = ymin
i1 = 0
for j1 in range(nt):
for k1 in range(nt):
diff2 = (tracersSub[i1, 2] - tracersSub[i1, 0]
)**2 + (tracersSub[i1, 3] -
tracersSub[i1, 1])**2
if (diff2 < min2):
min2 = diff2
minx = xmin + j1 / (nt - 1.) * (xmax - xmin)
miny = ymin + k1 / (nt - 1.) * (ymax - ymin)
it1 += 1
# get fixed point from this starting position using Newton's method
#TODO:
dl = np.min(
var.dx, var.dy
) / 100. # step-size for calculating the Jacobian by finite differences
it = 0
# tracers used to find the fixed point
tracersNull = np.zeros((5, 4))
point = np.array([minx, miny])
while True:
# trace field lines at original point and for Jacobian:
# (second order seems to be enough)
xx = np.zeros((5, 3))
xx[0, :] = np.array([point[0], point[1], p.Oz])
xx[1, :] = np.array([point[0] - dl, point[1], p.Oz])
xx[2, :] = np.array([point[0] + dl, point[1], p.Oz])
xx[3, :] = np.array([point[0], point[1] - dl, p.Oz])
xx[4, :] = np.array([point[0], point[1] + dl, p.Oz])
for it1 in range(5):
s = pc.stream(vv,
p,
hMin=hMin,
hMax=hMax,
lMax=lMax,
tol=tol,
interpolation=interpolation,
integration=integration,
xx=xx[it1, :])
tracersNull[it1, :2] = xx[it1, :2]
tracersNull[it1, 2:] = s.tracers[s.sl - 1, 0:2]
# check function convergence
ff = np.zeros(2)
ff[0] = tracersNull[0, 2] - tracersNull[0, 0]
ff[1] = tracersNull[0, 3] - tracersNull[0, 1]
#TODO:
if (sum(abs(ff)) <= 1e-4):
fixedPoint = np.array([point[0], point[1]])
break
# compute the Jacobian
fjac = np.zeros((2, 2))
fjac[0, 0] = (
(tracersNull[2, 2] - tracersNull[2, 0]) -
(tracersNull[1, 2] - tracersNull[1, 0])) / 2. / dl
fjac[0, 1] = (
(tracersNull[4, 2] - tracersNull[4, 0]) -
(tracersNull[3, 2] - tracersNull[3, 0])) / 2. / dl
fjac[1, 0] = (
(tracersNull[2, 3] - tracersNull[2, 1]) -
(tracersNull[1, 3] - tracersNull[1, 1])) / 2. / dl
fjac[1, 1] = (
(tracersNull[4, 3] - tracersNull[4, 1]) -
(tracersNull[3, 3] - tracersNull[3, 1])) / 2. / dl
# invert the Jacobian
fjin = np.zeros((2, 2))
det = fjac[0, 0] * fjac[1, 1] - fjac[0, 1] * fjac[1, 0]
#TODO:
if (abs(det) < dl):
fixedPoint = point
break
fjin[0, 0] = fjac[1, 1]
fjin[1, 1] = fjac[0, 0]
fjin[0, 1] = -fjac[0, 1]
fjin[1, 0] = -fjac[1, 0]
fjin = fjin / det
dpoint = np.zeros(2)
dpoint[0] = -fjin[0, 0] * ff[0] - fjin[0, 1] * ff[1]
dpoint[1] = -fjin[1, 0] * ff[0] - fjin[1, 1] * ff[1]
point += dpoint
# check root convergence
#TODO:
if (sum(abs(dpoint)) < 1e-4):
fixedPoint = point
break
if (it > 20):
fixedPoint = point
print "warning: Newton did not converged"
break
it += 1
# check if fixed point lies inside the cell
if ((fixedPoint[0] < tracers[iy, ix, 0, 0])
or (fixedPoint[0] > tracers[iy, ix + 1, 0, 0])
or (fixedPoint[1] < tracers[iy, ix, 0, 1])
or (fixedPoint[1] > tracers[iy + 1, ix, 0, 1])):
print "warning: fixed point lies outside the cell"
else:
x.append(fixedPoint[0])
y.append(fixedPoint[1])
#q.append()
fidx += 1
queue.put((x, y, q, fidx, iproc))