U.S. Department Of Energy's nuclear engineering education research: highlights of recent and current research-II. 2. Advanced Finite Element Discretizations for High-Energy Ion Transport

An efficient multigroup model that accurately describes electronic energy loss straggling was recently presented for use in multigroup Monte Carlo and deterministic high-energy ion transport codes. The model preserves the mean energy loss per path length, using the continuous slowing down (CSD) approximation, and mean-squared energy loss per path length, using strictly down-scatter multigroup cross sections, and accurately captures the energy spectrum of an initially monoenergetic ion beam. However, the dominant CSD energy loss process coupled with the small (but non-negligible) straggling poses a significant challenge for deterministic numerical solution when incident beams are monoenergetic or have discontinuous energy spectra. Such spectra broaden very slowly with depth into the target material, and thus, the interior distributions display sharpness and discontinuities. Advanced space-energy discretization methods are consequently necessary to achieve numerical robustness. In this paper, we investigate finite element solutions to this problem using two general families of discontinuous trial functions, one linear and the other nonlinear. The two families have been numerically tested, and we show results for 1.7-GeV protons incident on a tungsten target. For benchmarking purposes, we have also performed Monte Carlo (MC) simulations. Figure 1 displays the spatially converged energy spectrum with 200 groups after the beam has penetrated two-thirds of the ion range (85 cm). We contrast results from linear (LD), bilinear (BLD), and quadratic (QD) discontinuous trail functions against those from exponential-linear (LE), exponential-bilinear (BLE), and exponential-quadratic (QE) discontinuous trail functions. It is clear that the linear and bilinear results from both families are grossly inaccurate, both showing unacceptably high numerical straggling when compared against the MC results. The nonlinear results are everywhere positive while the linear schemes display negativities in the tails, although these are highly damped. In contrast, the QD and QE results are in excellent agreement with MC results. The QD scheme is slightly negative at the higher energy tail, but the QE is uniformly positive. The cubic discontinuous (CD) scheme (not shown) differs only in that the negativities have a shorter 'wavelength'. The significant difference between the two families is the computational labor. The nonlinear schemes can be slower by factors exceeding 100 because of the Newton iterations, which vary between 10 and 30 depending on what part of the spectrum is being relaxed but are relatively independent of the order of the trial function. The cost of each iteration depends sensitively on the quadrature order, which further depends on the trial function shape. For instance, a quadrature order of 4 was sufficient for the smooth but inaccurate BLE solution, but a 12-point quadrature was necessary to Newton-converge the QE case. Finally, in Table I, we show the order of accuracy for the linear family, obtained for the energy integrated spatial dose profile for increasing number of spatial cells K. The trend to the expected asymptotic accuracy is evident: third order for BLD, fifth order for QD, and seventh order for CD. In conclusion, although much work remains in optimizing the computations, it is clear that very high order finite element schemes are necessary for monoenergetic charged particle beam transport. This will be particularly true in multiple spatial dimensions in the presence of forward-peaked scattering. (authors)