Cyclotron Beam Dynamics Analysis

Overview
Structure of the code
Algorithm

Algorithm

The Runge-Kutt fourth order method is applied to calculate motion of particles in the CBDA code. There are two solvers to do this: "Track" solver and "Bunch" solver. The integration time step specifies the accuracy of the calculation. The simulations in each of the blocks ("Injection line", "Inflector" etc.) can be performed in any combination of the blocks in sequence.

Optimization

The optimization algorithm starts with backtracking the referent particle from its extraction radius to the radius, located close to the central region. Further, the referent particle is backtracked to one of the opening of the inflector housing for different positions of the accelerating gaps, in order to place them in the optimal way. In these calculations one must check if the final referent particle energy, Einj, at the opening is below the maximal injection energy. We created a data base of the numerically calculated electric fields in the two types of accelerating gaps, with the posts and without them. They are shown in Figure 1.



Figure 1: Acceleration gaps configuration.

The optimal positions of the gaps are found by minimizing functional F = UZe – Wg, where Ze is the referent particle charge, U the amplitude of the accelerating voltage, and Wg the energy gain of the referent particle in the gap. Thus, we obtain the maximal energy gain in all the gaps (see Figure 2.)



Figure 2: Optimal position for the acceleration gaps.

Performance

The problem becomes even more complicated when the beam space charge effects should be taken into account. The solution of the problem was attempted via upgrade of the recently developed computer code CBDA based on the beam dynamics calculation by parallel computing architecture.

CBDA code used two parallel methods for the accelerate calculation: OpenMP (by CPU multi-core) and CUDA (by using GPU ).

*OpenMP

Figure 1 shows the spiral inflector structure. For this configuration was performed a tracking of bunch with taken into account space charge effect. Figures 2 shows emittances at the inflector entrance. Figure 3 shows emittances at the inflector exit. Calculation was produced by two methods: PP and PIC. Table 1 present results for different number of CPU core.



Figure 1: Spiral inflector structure.



Figure 2



Figure 3



Table 1

*CUDA

The main part of calculation was performed without application of the CPU, and only a video card with GPU of about 100-200 processors was used. An application of the video card with the GPU instead of multi-processor computer is very cost effective solution in this case.

At present moment was used GeForce 8800GTX card (128 streaming processors). Computational procedure was optimized by several main functions:

o Track – calculate particle motion by Runge-Kutte method;

o Losses – calculate particle losses on the geometry structure;

o Rho – calculate a charge density function for the PIC method;

o Poisson/FFT – calculate FFT by 3 1D FFT and solve Poisson equation for PIC method;

o E_SC – interpolation an electric field to a nodal of mesh.

Table 1 shows performance produced by GPU. Notice, that acceleration of E_SC depends from number of nodes. At figure 1 presented effect from space chare for the central region of cyclotron.

Table 2 shows performance vs number of particle in bunch. In this test a space charge effect does not used.



Table 2



Table 3



Figure4




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