# The Fast Multipole Method in the Differential Algebra Framework for the Calculation of 3D Space Charge Fields

### A Dissertation

In partial fulfillment of the requirements for a degree of *Doctor
of Philosophy* from Michigan State University.

### Abstract

The space charge effect is one of the most important collective effects in beam dynamics
studies. In many cases, numerical simulations are inevitable in order to get a clear understanding
of this effect. The particle-particle interaction algorithms and the particle-in-cell
algorithms are widely used in space charge effect simulations. But they both have difficulties
in dealing with highly correlated beams with abnormal distributions or complicated geometries.
We developed a new algorithm to calculate the three dimensional self-field between
charged particles by combining the differential algebra (DA) techniques with the fast multipole
method (FMM). The FMM hierarchically decomposes the whole charged domain into
many small regions. For each region it uses multipole expansions to represent the potential/field
contributions from the particles far away from the region and then converts the
multipole expansions into a local expansion inside the region. The potential/field due to the
far away particles is calculated from the expansions and the potential/field due to the nearby
particles is calculated from the Coulomb force law. The DA techniques are used in the calculation,
translation and converting of the expansions. The new algorithm scales linearly with
the total number of particles and it is suitable for any arbitrary charge distribution. Using
the DA techniques, we can calculate both the potential/field and its high order derivatives,
which will be useful for the purpose of including the space charge effect into transfer maps
in the future.
We first present the single level FMM, which decomposes the whole domain into boxes of
the same size. It works best for charge distributions that are not overly non-uniform. Then
we present the multilevel fast multipole algorithm (MLFMA), which decomposes the whole
domain into different sized boxes according to the charge density. Finer boxes are generated
where the higher charge density exists; thus the algorithm works for any arbitrary charge
distribution. A Message Passing Interface (MPI) based parallel version of the MLFMA is
developed, so that we can take advantage of cluster machines and enhance our simulation
ability. The algorithms are described in details and the numerical experimental results about
the efficiency and accuracy of the algorithm are presented and discussed. In the end, we
give an example of using this algorithm in the photo emission process simulation. Some
simulation related topics are discussed, such as: how to choose the proper units for the
variables in the beam dynamics equations, how to transform the space charge fields from
the bunch frame to the laboratory frame, and how to avoid artificial collisions between the
charged particles.

H. Zhang (2013)

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