Applications of Parallel Computational Methods to Charged-Particle Beam Dynamics Page: 3 of 5
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map affects beam lifetime due to incoherent resonances or
diffusion processes. To study these effects, we have
developed a C + + code DUMBBB for fast tracking of
single particles in the presence of weak-strong beam-beam
interactions (for both parasitic and design interactions, i.e.
off- or on-center). Being a single-particle code, paralleliza-
tion reduces to the task of running many instances of the
same code acting on different parts of a huge particle
ensemble; communications is only required for calculation
of collective quantities such as particle loss rates or
emittances.
The code models the beam-beam interactions as a
synchro-betatron mapping [5], the beam-beam kick itself is
calculated from the Bassetti-Erskine [6] formula, using the
Chiarella-Matta-Reichel approximation [7] for the evalua-
tion of the complex error function. In a hadron machine, it
is important to avoid all sources of numerical noise; the
Chiarella algorithm is implemented as a templated C + +
function with accuracy selectable at compile time. We find
a 10-6 relative accuracy sufficient for turn numbers in the
105 range.
The weak-beam part of the machine is modeled by a
concatenation of beam-beam elements, linear 6 x 6 trans-
fer maps between non-linear elements (obtained by having
a Perl script run MAD8 on a optics description file), a
noise-inducing element to model emittance growth due to
scattering processes, and a energy-dependent tune advance
element to introduce total ring chromaticity.
The code allows for full coupling. All element transfer
functions are templated with respect to the type of phase-
space variables; in particular, they can operate on
differential-algebraic quantities. This allows for finding
exact solutions for the linear part of the one-turn map at
start-up and constructing invariant initial weak and strong
6 x 6 distributions, matched to measured emittances.
Beam-beam elements have specialized functions depending
on whether or not the strong beam shows hourglass effect,
tilting, or position-depending tilting during an interaction.
Also, they are templatized with respect to the number of
slices used in the synchro-betatron mapping.
The aggregated lattice is repeatedly applied to real-value
phase-space vectors of the initial weak distribution, which
can be "de-cored" to remove particles from the core of the
distribution, which are not expected to contribute to
diffusive or resonant particle losses. Care must be taken
to keep this operation invariant with respect to the one-
turn map. The particles' excursions in action-angle space
are recorded; once every few thousand turns, the Jx, Jy
space is swept and particles beyond a certain action
aperture are counted. This way, we obtain a plot of
particle loss vs. time for different assumptions about the
limiting aperture of the machine. We typically run 1010
particle turns. The resulting dependencies are fitted with
respect to i against exp(-t/z) and exp(- t/z) particle loss
behaviors, which are the limiting cases of solutions of the
diffusion equation with absorbing boundary conditions for
small and large-aperture boundaries. Due to the uncertain-7
6
54
2
w
:a3
2
1Pbar bunch 1 lifetime at 150 GeV (A. Kabel)
chrom 0
- - - - - - - - - - - - 4 - - - - - - - - - - -
' ctrom -
Lifetime from N No exp[ t/t]
chromy 4
-
cThrom d.
chrom- 90 1
3 4 5 6 7 8 9 10
Vertical aperture [o]
Fig. 3. Lifetime vs. vertical aperture and vertical chromaticity for
Tevatron at injection.
ties of the diffusion model, the real aperture, and the
simplifications in the model, the resulting lifetime should
not be viewed as an absolute prediction, but as a figure of
merit establishing signatures of the real lifetime of the
machine.
We have done a series of parameter studies for the
Tevatron at injection (150 GeV, 72 parasitic crossings,
modeled as single-slice interactions). A typical result for
varying chromaticity is shown in Fig. 3. Other parameter
studies included sweeps of helix separations; weak-beam
emittances, strong-beam charges, and two different bunch
train schemes for 18 bunches on each of the bunch trains,
resulting in lifetime differences of factors of two depending
on deleting the odd- or even-numbered interactions; the
latter result was checked independently with resonance
strength studies.
To model non-linear effects due to the lattice, we have
implemented a method for high-speed evaluation of
multivariate polynomials. The method relies on the
recursive definition of Pn, a v-variate homogeneous
polynomial of degree n, as a direct sum Pn = P_1 P,1
and a recursive evaluation algorithm Pn(x1 ... , xv) =
''x) +P2(x2,... , xj). Using C + +'s tem-
plated data structure mechanisms for the definition of P
and inlining for the definition of the polynomial evalua-
tion, the method effectively generates explicit expressions
for Horner's scheme for any order at compile time. The
method is easily generalized to inhomogeneous polyno-
mials. We observe floating point efficiencies of >0.85 on
Intel hardware and a speed gain of a factor of 4 as
compared to standard implementations; still, we would
need to gain another factor of 10 in speed to use 10th
order polynomial transfer maps between beam-beam
interactions.
3. The strong-strong beam-beam effect: NIMZOVICH
In the strong-strong realm, the colliding bunches
influence each other substantially. Little is known analy-
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Kabel, A.; Cai, Y.; Dohlus, M.; Sen, T. & Uplenchwar, R. Applications of Parallel Computational Methods to Charged-Particle Beam Dynamics, article, October 16, 2007; [Menlo Park, California]. (https://digital.library.unt.edu/ark:/67531/metadc890380/m1/3/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.