Abstract
High power sources of electromagnetic energy often require complicated structures to support electromagnetic modes and shape electromagnetic fields to maximize the coupling of the field energy to intense relativistic electron beams. Geometric fidelity is critical to the accurate simulation of these High Power Electromagnetic (HPEM) sources. Here, we present a fast and geometrically flexible approach to calculate the solution to Maxwell’s equations in vector potential form under the Lorenz gauge. The scheme is an implicit, linear-time, high-order, A-stable method that is based on the method of lines transpose (MOLT). As presented, the method is fourth order in time and second order in space, but the A-stable formulation could be extended to both high order in time and space. An O(n) fast convolution is employed for space-integration. The main focus of this work is to develop an approach to impose perfectly electrically conducting (PEC) boundary conditions in MOLT by extending our past work on embedded boundary methods. As the method is A-stable, it does not suffer from small time step limitations that are found in explicit finite difference time domain methods when using either embedded boundary or cut-cell methods to capture geometry. This is a major advance for the simulation of HPEM devices. While there is no conceptual limitation to develop this in 3D, our initial work has centered on 2D. The extension to 3D requires validation that the proposed fixed point iteration will converge and is the subject of our follow-up work. The eventual goal is to combine this method with particle methods for the simulations of plasma. In the current work, the scheme is evaluated for EM wave propagation within an object that is bounded by PEC. The consistency and performance of the scheme are confirmed using the ping test and frequency mode analysis for rotated square cavities—a standard test in the HPEM community. We then demonstrate the diffraction Q value test and the use of this method for simulating an A6 magnetron. The ability to handle both PEC and open boundaries in a standard device test problem, such as the A6, gives confidence on the robustness of this new method.
Highlights
The generation of coherent, high power electromagnetic (HPEM) radiation via the interaction of vacuum electromagnetic fields and intense relativistic electron beams continues to remain an active area of research, despite tracing its roots to the days of Hertz and Marconi and the intensive development of radar during the Second World War
These standard Particle In Cell (PIC) methods often require highly parallel implementation to reduce the effect of the structure ortho-normal grid and/or so-called “cut cells” where the geometry is approximated by linear interpolation through the cells
The alternating direction implicit (ADI)-finite difference time domain (FDTD) method combined with the Dey–Mittra embedded boundary method can model the curved domains associated with complex structures and time step sizes beyond the CFL limit
Summary
The generation of coherent, high power electromagnetic (HPEM) radiation via the interaction of vacuum electromagnetic fields and intense relativistic electron beams continues to remain an active area of research, despite tracing its roots to the days of Hertz and Marconi and the intensive development of radar during the Second World War. The ADI-FDTD method combined with the Dey–Mittra embedded boundary method can model the curved domains associated with complex structures and time step sizes beyond the CFL limit.5 The efficiency of this method depends on the one-dimensional tridiagonal solvers used underneath and that will cause a major bottleneck issue and affects the scalability of the scheme. By imposing the Lorenz gauge condition, a wave equation for the scalar potential φ in free space is obtained,
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