Fast hybrid particle-in-cell technique for pulsed-power accelerators

Hybrid-implicit particle-in-cell (PIC) algorithms permit the simulation of complex problems involving both kinetic and fluid plasma regimes over large spatial and temporal scales. Fluid electrons can be computationally fast where and when fluid assumptions are valid. Additional flexibility is obtained if discrete PIC macroparticles, with velocities advanced by either fluid or kinetic equations, are permitted to dynamically migrate between the two descriptions based on phase space criteria. Ideally, these migrations result in energetic particles treated kinetically and dense thermal plasma particles as a fluid. With an energy-conserving particle advance, resolution of the plasma Debye length is not required for numerical accuracy or stability. For pulsed-power applications, the simulation time step is usually constrained by the electron cyclotron frequency, not the more restrictive plasma frequency. A new implicit technique permits accurate particle orbits even at highly underresolved cyclotron frequencies. Thus, greater temporal and spatial scales can be accurately modeled relative to conventional PIC techniques. In this paper, we describe the hybrid PIC technique and fully electromagnetic, hybrid simulations of plasma evolution and current shunting in an idealized accelerator designed for driving a $Z$-pinch load. The dynamics of electrode heating, electron transport, and surface contaminant evolution are studied in a series of relativistic hybrid-implicit PIC simulations. These dynamics can lead to the shunting of current before reaching the $Z$-pinch load, thus degrading load performance. Examining two previously published power flow problems, we compare results from fully kinetic, multifluid, and hybrid kinetic-fluid simulations and discuss the computational performance of these three options. The key thrust of the work is to identify possible computational acceleration, through hybrid methods, required for accelerator understanding and design.

Figure 1

The positions of the distribution moments ($\rho$, $N$, $v$, $P$, $U$, and $T$), electromagnetic fields, and currents $J$ are plotted on a representative 2D rectangular mesh in ($x$, $y$) Cartesian geometry with $I$ and $j$ indices. Also plotted is a representative particle carrying attributes which are bilinearly interpolated to the full and edge position on the grid. Changes in internal energy and momenta are then interpolated back to particles.

Figure 2

For a charged particle with $r_L=1$ and $v_x$ $(o)=r_L{\omega }_c$, the charge particle motion in the $x-y$ plane is shown for the exact solution (red line) and MI calculation ($+$) with ${\omega }_c\mathrm{\Delta }t=10\pi$.

Figure 3

For a charged particle with $r_L=1$ and $v_x$ $(o)=0.2c$, $v_{ExB}=0.1c$, the charge particle motion in the $X\text{−}Y$ plane is shown for the exact solution (red line) and MI calculation ($+$) with ${\omega }_c\mathrm{\Delta }t=5\pi$.

Figure 4

A schematic of the 3D simulation is shown with the $Z$ circuit input and a short-circuit load.

Figure 5

The electron density is plotted 90 ns into the four simulations in the $\theta =0$ plane (center of post). The plot shows results from (a) all kinetic particles, (b) the multifluid or all fluid particles, (c) all PMH with fluid particles transitioning to kinetic above 50 keV, and (d) fluid electrons with PMH ions. All these simulations have dynamic time steps such that ${\omega }_c\mathrm{\Delta }t<9$.

Figure 6

The (a) load current calculated at $z=6\mathrm{cm}$ and (b) loss current (difference between total current entering the MITLs and the load current) from four simulations (kinetic, multifluid, PMH, and fluid $e$ with PMH ions) using the geometry in Fig. 4 are shown.

Figure 7

The electron density is plotted 120 ns into the four 2D inner MITL simulations. The plot shows results from (a) all kinetic particles, (b) the multifluid particles, (c) all PMH with fluid particles transitioning to kinetic above 50 keV, and (d) fluid electrons with PMH ions. All these simulations have dynamic time steps such that ${\omega }_c\mathrm{\Delta }t<9$.

Figure 8

The calculated current loss for four inner MITL simulations (kinetic, multifluid, PMH, and fluid $e$ with PMH ions). The drive current rises to 22.6 MA by 120 ns.