Note
Go to the end to download the full example code
Automatic Growth Rates Re-Computation
This example shows how to use the the auto recomputing functionality for growth rates in the analytical classes. To follow this example please first have a look at the Analytical Nagaitsev example as well as the relevant FAQ section.
This script will showcase the functionality by following an identical flow to the Nagaitsev example linked above and expanding on the relevant parts. Demonstration will be done using the SPS top protons line.
import logging
from dataclasses import dataclass
import matplotlib.pyplot as plt
import numpy as np
import xpart as xp
import xtrack as xt
from xibs.analytical import NagaitsevIBS
from xibs.formulary import _bunch_length, _geom_epsx, _geom_epsy, _sigma_delta, _percent_change
from xibs.inputs import BeamParameters, OpticsParameters
logging.basicConfig(
level=logging.WARNING,
format="[%(asctime)s] [%(levelname)s] - %(module)s.%(funcName)s:%(lineno)d - %(message)s",
datefmt="%H:%M:%S",
)
plt.rcParams.update(
{
"font.family": "serif",
"font.size": 20,
"axes.titlesize": 20,
"axes.labelsize": 20,
"xtick.labelsize": 20,
"ytick.labelsize": 20,
"legend.fontsize": 15,
"figure.titlesize": 20,
}
)
Let’s start by defining the line and particle information, as well as some parameters for later use:
Setting up line and particles
Let’s start by loading the xtrack.Line, activating RF cavities and
generating a matched Gaussian bunch:
line = xt.Line.from_json(line_file)
line.build_tracker()
line.optimize_for_tracking()
twiss = line.twiss(method="4d")
# ----- Power accelerating cavities ----- #
rf_voltage = 4 # in MV
rf_frequency = 200e6
harmonic_number = 4653
cavity_name = "actcse.31632"
cavity_lag = 180
line[cavity_name].voltage = rf_voltage * 1e6 # to be given in [V] here
line[cavity_name].lag = cavity_lag
line[cavity_name].frequency = rf_frequency
particles = xp.generate_matched_gaussian_bunch(
num_particles=n_part,
total_intensity_particles=bunch_intensity,
nemitt_x=nemitt_x,
nemitt_y=nemitt_y,
sigma_z=sigma_z,
particle_ref=line.particle_ref,
line=line,
)
Loading line from dict: 0%| | 0/9866 [00:00<?, ?it/s]
Loading line from dict: 28%|██▊ | 2787/9866 [00:00<00:00, 27862.52it/s]
Loading line from dict: 56%|█████▋ | 5574/9866 [00:00<00:00, 27749.46it/s]
Loading line from dict: 85%|████████▌ | 8389/9866 [00:00<00:00, 27930.02it/s]
Loading line from dict: 100%|██████████| 9866/9866 [00:00<00:00, 27922.93it/s]
Done loading line from dict.
[07:28:27] [WARNING] - kernel_definitions.<module>:221 - Xcoll not installed, skipping collimator elements
Compiling ContextCpu kernels...
Done compiling ContextCpu kernels.
Disable xdeps expressions
Remove markers
Remove inactive multipoles
Merge consecutive multipoles
Remove redundant apertures
Remove zero length drifts
Merge consecutive drifts
Use simple bends
Use simple quadrupoles
Rebuild tracker data
Compiling ContextCpu kernels...
Done compiling ContextCpu kernels.
Compiling ContextCpu kernels...
Done compiling ContextCpu kernels.
*** Maximum RMS bunch length 0.23625521889289555m.
/opt/hostedtoolcache/Python/3.11.9/x64/lib/python3.11/site-packages/scipy/integrate/_quadpack_py.py:1272: IntegrationWarning: The integral is probably divergent, or slowly convergent.
quad_r = quad(f, low, high, args=args, full_output=self.full_output,
... distance to target bunch length: -1.0000e-01
... distance to target bunch length: 1.2973e-01
... distance to target bunch length: 9.7855e-02
... distance to target bunch length: 1.1051e-02
... distance to target bunch length: -2.6587e-03
... distance to target bunch length: -4.0900e-05
... distance to target bunch length: 1.5139e-08
... distance to target bunch length: -5.4040e-07
--> Bunch length: 0.10000001513917059
--> Emittance: 0.3014753866154415
We can compute initial (geometrical) emittances as well as the bunch length
from the xtrack.Particles object:
geom_epsx = _geom_epsx(particles, twiss.betx[0], twiss.dx[0])
geom_epsy = _geom_epsy(particles, twiss.bety[0], twiss.dy[0])
bunch_l = _bunch_length(particles)
sig_delta = _sigma_delta(particles)
Instantiate Analytical IBS Objects
Let us instantiate three instances of the NagaitsevIBS class,
where each will be used for a different updating behaviour of the growth rates.
beam_params = BeamParameters(particles)
optics = OpticsParameters(twiss)
IBS = NagaitsevIBS(beam_params, optics) # will recompute rates at a given frequency
AUTO_IBS = NagaitsevIBS(beam_params, optics) # will auto-recompute rates only
Let’s get the growth rates and see the evolution of the transverse emittances, \(\sigma_{\delta}\) and bunch length. We can then have a look at the relative change from one second to the next.
IBS.growth_rates(geom_epsx, geom_epsy, sig_delta, bunch_l)
AUTO_IBS.growth_rates(geom_epsx, geom_epsy, sig_delta, bunch_l)
new_geom_epsx, new_geom_epsy, new_sig_delta, new_bunch_length = IBS.emittance_evolution(
geom_epsx, geom_epsy, sig_delta, bunch_l
)
# Let's see the relative change
print(f"Geom. epsx: {geom_epsx:.2e} -> {new_geom_epsx:.2e} | ({100 * _percent_change(geom_epsx, new_geom_epsx):.2e}% change)")
print(f"Geom. epsy: {geom_epsy:.2e} -> {new_geom_epsy:.2e} | ({100 * _percent_change(geom_epsy, new_geom_epsy):.2e}% change)")
print(f"Sigma delta: {sig_delta:.2e} -> {new_sig_delta:.2e} | ({100 * _percent_change(sig_delta, new_sig_delta):.2e}% change)")
print(f"Bunch length: {bunch_l:.2e} -> {new_bunch_length:.2e} | ({100 * _percent_change(bunch_l, new_bunch_length):.2e}% change)")
Geom. epsx: 2.05e-09 -> 2.05e-09 | (9.65e-04% change)
Geom. epsy: 2.13e-09 -> 2.13e-09 | (-7.92e-07% change)
Sigma delta: 1.56e-04 -> 1.56e-04 | (3.95e-04% change)
Bunch length: 9.95e-02 -> 9.95e-02 | (3.95e-04% change)
Preparing for Simulation of Evolution
We will loop over time steps (seconds in this case) for 5 hours equivalent time, and specify the auto-recomputing in these steps where relevant. Let’s set up the utilities we will need for this:
nsecs = 5 * 3_600 # that's 5h
ibs_step = 10 * 60 # fixed interval for rates recomputing
seconds = np.linspace(0, nsecs, nsecs, dtype=int)
# This is the threshold that would trigger the auto-recomping of the growth rates.
AUTO_PERCENT = 15 # threshold: 15% change from last growth rates update
# Set up a dataclass to store the results
@dataclass
class Records:
"""Dataclass to store (and update) important values through tracking."""
epsilon_x: np.ndarray # geometric horizontal emittance in [m]
epsilon_y: np.ndarray # geometric vertical emittance in [m]
sig_delta: np.ndarray # momentum spread
bunch_length: np.ndarray # bunch length in [m]
epsx_rel_to_last_ref: np.ndarray # relative change of epsx to last reference
epsy_rel_to_last_ref: np.ndarray # relative change of epsy to last reference
sigd_rel_to_last_ref: np.ndarray # relative change of sigd to last reference
bl_rel_to_last_ref: np.ndarray # relative change of bl to last reference
@classmethod
def init_zeroes(cls, n_turns: int):
return cls(
epsilon_x=np.zeros(n_turns, dtype=float),
epsilon_y=np.zeros(n_turns, dtype=float),
sig_delta=np.zeros(n_turns, dtype=float),
bunch_length=np.zeros(n_turns, dtype=float),
epsx_rel_to_last_ref=np.zeros(n_turns, dtype=float),
epsy_rel_to_last_ref=np.zeros(n_turns, dtype=float),
sigd_rel_to_last_ref=np.zeros(n_turns, dtype=float),
bl_rel_to_last_ref=np.zeros(n_turns, dtype=float),
)
def update_at_turn(self, turn: int, epsx: float, epsy: float, sigd: float, bl: float, epsxrel: float = 0.0, epsyrel: float = 0.0, sigdrel: float = 0.0, blrel: float = 0.0):
"""Works for turns / seconds, just needs the correct index to store in."""
self.epsilon_x[turn] = epsx
self.epsilon_y[turn] = epsy
self.sig_delta[turn] = sigd
self.bunch_length[turn] = bl
self.epsx_rel_to_last_ref[turn] = epsxrel
self.epsy_rel_to_last_ref[turn] = epsyrel
self.sigd_rel_to_last_ref[turn] = sigdrel
self.bl_rel_to_last_ref[turn] = blrel
# Initialize the dataclass & store the initial values
regular = Records.init_zeroes(nsecs)
auto = Records.init_zeroes(nsecs)
regular.update_at_turn(0, geom_epsx, geom_epsy, sig_delta, bunch_l)
auto.update_at_turn(0, geom_epsx, geom_epsy, sig_delta, bunch_l)
# These arrays we will use to see when the auto-recomputing kicks in
auto_recomputes = np.zeros(nsecs, dtype=float)
fixed_recomputes = np.zeros(nsecs, dtype=float)
Analytical Evolution Over Time
Let’s now loop and specify the auto-recomputing where relevant.
for sec in range(1, nsecs):
# This is not necessary, just for showcasing in this tutorial
n_recomputes_for_auto = AUTO_IBS._number_of_growth_rates_computations
# ----- Potentially re-compute the IBS growth rates ----- #
if (sec % ibs_step == 0) or (sec == 1):
print(f"At {sec}s: Fixed interval recomputing of the growth rates")
# Below always re-computes the rates every 'ibs_step' seconds
IBS.growth_rates(
regular.epsilon_x[sec - 1],
regular.epsilon_y[sec - 1],
regular.sig_delta[sec - 1],
regular.bunch_length[sec - 1],
)
# ----- Compute the new emittances ----- #
new_emit_x, new_emit_y, new_sig_delta, new_bunch_length = IBS.emittance_evolution(
epsx=regular.epsilon_x[sec - 1],
epsy=regular.epsilon_y[sec - 1],
sigma_delta=regular.sig_delta[sec - 1],
bunch_length=regular.bunch_length[sec - 1],
dt=1.0, # get at next second
)
regular.update_at_turn(sec,
new_emit_x,
new_emit_y,
new_sig_delta,
new_bunch_length,
_percent_change(IBS._refs.epsx, new_emit_x),
_percent_change(IBS._refs.epsy, new_emit_y),
_percent_change(IBS._refs.sigma_delta, new_sig_delta),
_percent_change(IBS._refs.bunch_length, new_bunch_length),
)
# ----- Potentially auto-update growth rates and compute new emittances ----- #
# The AUTO_IBS is given our threshold for auto-recomputing and will decide to update
# its growth rates itself if any of the quantities change by more than 15%.
anew_emit_x, anew_emit_y, anew_sig_delta, anew_bunch_length = AUTO_IBS.emittance_evolution(
epsx=auto.epsilon_x[sec - 1],
epsy=auto.epsilon_y[sec - 1],
sigma_delta=auto.sig_delta[sec - 1],
bunch_length=auto.bunch_length[sec - 1],
dt=1.0, # get at next second
auto_recompute_rates_percent=AUTO_PERCENT,
)
auto.update_at_turn(
sec,
anew_emit_x,
anew_emit_y,
anew_sig_delta,
anew_bunch_length,
_percent_change(AUTO_IBS._refs.epsx, anew_emit_x),
_percent_change(AUTO_IBS._refs.epsy, anew_emit_y),
_percent_change(AUTO_IBS._refs.sigma_delta, anew_sig_delta),
_percent_change(AUTO_IBS._refs.bunch_length, anew_bunch_length),
)
# ----- Check if the rates were auto-recomputed ----- #
# This is also not necessary, just for showcasing in this tutorial
if AUTO_IBS._number_of_growth_rates_computations > n_recomputes_for_auto:
print(f"At {sec}s - Auto re-computed growth rates")
auto_recomputes[sec] = 1
# Here we also aggregate the fixed recomputes
for sec in seconds:
if (sec % ibs_step == 0) or (sec == 1):
fixed_recomputes[sec - 1] = 1
# And these are simply 1D arrays of the seconds at which re-computing happened
where_auto_recomputes = np.flatnonzero(auto_recomputes)
where_fixed_recomputes = np.flatnonzero(fixed_recomputes)
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At 1200s: Fixed interval recomputing of the growth rates
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At 1800s: Fixed interval recomputing of the growth rates
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At 1998s - Auto re-computed growth rates
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At 2010s - Auto re-computed growth rates
At 2016s - Auto re-computed growth rates
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At 2400s: Fixed interval recomputing of the growth rates
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At 3000s: Fixed interval recomputing of the growth rates
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At 3600s: Fixed interval recomputing of the growth rates
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At 4200s: Fixed interval recomputing of the growth rates
At 4209s - Auto re-computed growth rates
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At 4800s: Fixed interval recomputing of the growth rates
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At 5381s - Auto re-computed growth rates
At 5399s - Auto re-computed growth rates
At 5400s: Fixed interval recomputing of the growth rates
At 5417s - Auto re-computed growth rates
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At 6000s: Fixed interval recomputing of the growth rates
At 6019s - Auto re-computed growth rates
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At 6600s: Fixed interval recomputing of the growth rates
At 6601s - Auto re-computed growth rates
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At 7200s: Fixed interval recomputing of the growth rates
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At 7800s: Fixed interval recomputing of the growth rates
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At 8400s: Fixed interval recomputing of the growth rates
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At 9000s: Fixed interval recomputing of the growth rates
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At 9600s: Fixed interval recomputing of the growth rates
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At 10200s: Fixed interval recomputing of the growth rates
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At 10800s: Fixed interval recomputing of the growth rates
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Let’s see how many recomputes happened in total for each scenario:
print(f"Fixed re-computes: {IBS._number_of_growth_rates_computations}")
print(f"Auto re-computes: {AUTO_IBS._number_of_growth_rates_computations}")
Fixed re-computes: 31
Auto re-computes: 1621
That’s almost 1600 additional updates of the growth rates that were deemed necessary by the auto-recomputing mechanism. Let’s see the effect it has had on our results by having a look at the evolution of emittances over time:
fig, axs = plt.subplot_mosaic([["epsx", "epsy"], ["sigd", "bl"]], sharex=True, figsize=(13, 8.5))
# We will add vertical lines at the times where recomputing of the growth
# rates happened (do this first so they show up in the background)
for axis in axs.values():
for sec in where_fixed_recomputes:
axis.axvline(sec / 3600, color="C0", linestyle="--", lw=1, alpha=0.5)
for sec in where_auto_recomputes:
axis.axvline(sec / 3600, color="C1", linestyle="-", alpha=0.035)
axs["epsx"].plot(seconds / 3600, 1e9 * regular.epsilon_x, lw=2, label=f"Fixed ({int(ibs_step / 60)} mins)")
axs["epsy"].plot(seconds / 3600, 1e9 * regular.epsilon_y, lw=2, label=f"Fixed ({int(ibs_step / 60)} mins)")
axs["sigd"].plot(seconds / 3600, 1e4 * regular.sig_delta, lw=2, label=f"Fixed ({int(ibs_step / 60)} mins)")
axs["bl"].plot(seconds / 3600, 1e2 * regular.bunch_length, lw=2, label=f"Fixed ({int(ibs_step / 60)} mins)")
axs["epsx"].plot(seconds / 3600, 1e9 * auto.epsilon_x, lw=2, label=f"Auto ({AUTO_PERCENT:.0f}% change)")
axs["epsy"].plot(seconds / 3600, 1e9 * auto.epsilon_y, lw=2, label=f"Auto ({AUTO_PERCENT:.0f}% change)")
axs["sigd"].plot(seconds / 3600, 1e4 * auto.sig_delta, lw=2, label=f"Auto ({AUTO_PERCENT:.0f}% change)")
axs["bl"].plot(seconds / 3600, 1e2 * auto.bunch_length, lw=2, label=f"Auto ({AUTO_PERCENT:.0f}% change)")
# Axes parameters
axs["epsx"].set_ylabel(r"$\varepsilon_x$ [$10^{-9}$m]")
axs["epsy"].set_ylabel(r"$\varepsilon_y$ [$10^{-9}$m]")
axs["sigd"].set_ylabel(r"$\sigma_{\delta}$ [$10^{-4}$]")
axs["bl"].set_ylabel(r"Bunch length [cm]")
for axis in (axs["epsy"], axs["bl"]):
axis.yaxis.set_label_position("right")
axis.yaxis.tick_right()
for axis in (axs["sigd"], axs["bl"]):
axis.set_xlabel("Duration [h]")
fig.align_ylabels((axs["epsx"], axs["sigd"]))
fig.align_ylabels((axs["epsy"], axs["bl"]))
fig.suptitle("Analytical Evolution of Emittances from IBS\nSPS Top Protons")
plt.legend(title="Recompute Rates")
plt.tight_layout()
plt.show()
Let’s keep in mind that the formula for the evolution of these properties from IBS is an exponential that depends on the value at the previous step, and the growth rate for the given plane. As long as the growth rate is not re-computed, then the relative change from one time step to the next will be the same. We can have a look at the relative change from the last time growth rates were updated:
fig, axs = plt.subplot_mosaic([["epsx", "epsy"], ["sigd", "bl"]], sharex=True, figsize=(13, 8.5))
# We will add vertical lines at the times where recomputing of the growth
# rates happened (do this first so they show up in the background)
for axis in axs.values():
for sec in where_fixed_recomputes:
axis.axvline(sec / 3600, color="C0", linestyle="--", lw=1, alpha=0.5)
for sec in where_auto_recomputes:
axis.axvline(sec / 3600, color="C1", linestyle="-", alpha=0.035)
axs["epsx"].plot(seconds / 3600, 1e2 * regular.epsx_rel_to_last_ref, lw=2, label=f"Fixed ({int(ibs_step / 60)} mins)")
axs["epsy"].plot(seconds / 3600, 1e2 * regular.epsy_rel_to_last_ref, lw=2, label=f"Fixed ({int(ibs_step / 60)} mins)")
axs["sigd"].plot(seconds / 3600, 1e2 * regular.sigd_rel_to_last_ref, lw=2, label=f"Fixed ({int(ibs_step / 60)} mins)")
axs["bl"].plot(seconds / 3600, 1e2 * regular.bl_rel_to_last_ref, lw=2, label=f"Fixed ({int(ibs_step / 60)} mins)")
axs["epsx"].plot(seconds / 3600, 1e2 * auto.epsx_rel_to_last_ref, lw=2, label=f"Auto ({AUTO_PERCENT:.0f}% change)")
axs["epsy"].plot(seconds / 3600, 1e2 * auto.epsy_rel_to_last_ref, lw=2, label=f"Auto ({AUTO_PERCENT:.0f}% change)")
axs["sigd"].plot(seconds / 3600, 1e2 * auto.sigd_rel_to_last_ref, lw=2, label=f"Auto ({AUTO_PERCENT:.0f}% change)")
axs["bl"].plot(seconds / 3600, 1e2 * auto.bl_rel_to_last_ref, lw=2, label=f"Auto ({AUTO_PERCENT:.0f}% change)")
# Axes parameters
axs["epsx"].set_ylabel(r"$\varepsilon_x$")
axs["epsy"].set_ylabel(r"$\varepsilon_y$")
axs["sigd"].set_ylabel(r"$\sigma_{\delta}$")
axs["bl"].set_ylabel(r"Bunch length")
for axis in (axs["epsy"], axs["bl"]):
axis.yaxis.set_label_position("right")
axis.yaxis.tick_right()
for axis in (axs["sigd"], axs["bl"]):
axis.set_xlabel("Duration [h]")
for axis in axs.values():
axis.axhline(AUTO_PERCENT, color="black", linestyle="--", alpha=0.5, label="Threshold")
axis.yaxis.set_major_locator(plt.MaxNLocator(3))
axis.set_yscale("log")
fig.align_ylabels((axs["epsx"], axs["sigd"]))
fig.align_ylabels((axs["epsy"], axs["bl"]))
fig.suptitle("Percent change from values at previous growth rate update\nSPS Top Protons")
plt.legend(title="Recompute Rates")
plt.tight_layout()
plt.show()
We can see from this plot that the relative change for the “fixed” interval scenario keeps increasing between re-computes of the growth rates. Because we pushed the beam parameters to the extreme, the emittances grow “too fast” from the start to the first recompute of the rates 10 minutes in.
The “auto-only” recomputing scenario initially updates its growth rates very frequently in the beginning of the simulation: almost every time step for the first 10 minutes of simulated beam time. As the horizontal emittance and the \(\delta_p\) and bunch length grow, the effect of IBS decreases and the rates are not updated as frequently.
Takeaways
One can see the difference made by the auto-recomputing, especially in the horizontal plane where a large gap to the “fixed interval” scenario is observed for the final values reached, despite an asymptotic behaviour. In the beginning of the simulation, auto-recomputing is very frequent and avoid an unrealistic growth of the emittances.
It is important for the user to choose an appropriate threshold value. A value too high will lead to the rates not updating frequently enough and potentially unrealistic evolutions, while a value too low will lead to the rates re-computing at every step for a long time, which is computationally expensive and unnecessary.
References
The use of the following functions, methods, classes and modules is shown in this example:
analytical:NagaitsevIBS,growth_rates,integrals,emittance_evolutioninputs:BeamParameters,OpticsParameters
Total running time of the script: (2 minutes 32.134 seconds)