Doppler broadening using discrete cosine transform and kernel reconstruction for spatially variable media

Abstract

Doppler broadening plays key roles in understanding the temperature dependence of materials’ particle reaction rates, which are vital for high fidelity nuclear reactor physics simulations. However, the treatment of spatially continuously variable temperature distribution is far from its maturity to reach the goal of routine high fidelity reactor whole core transport. This article made an attempt to move one step closer to the goal by proposing and verifying a solution package innovative in three aspects. First, we present the DCT-DB Doppler broadening method, efficient in computation and compact for storage, being based on discrete cosine transform by reformulating the Doppler broadening problem as solving a heat equation. The total, elastic, absorption, capture and fission resonance plus background cross sections (if needed) of 512 nuclides from the most recent ENDF/B-VIII.0 data library are studied in the temperature ranging from 200 K to 3,000 K. The maximum relative errors achieve the 0.001 target (in the meaningful range) after validated against NJOY 2016. The new method is computationally efficient, where for the nuclides with maximum intermediate energy grid size of 226 (more than 67 millions), GPU delivers a computing power equivalent to 1 to 2 orders of magnitudes of CPU cores running with the NJOY Doppler broadening method, while generating cross sections at 1 to 2 orders of magnitudes lower errors. The storage of coefficients for all 512 nuclides compresses in an almost numerically lossless way to be 715 MB. Second, we combine the kernel reconstruction method with improved Monte Carlo delta tracking and track length (as track integral) estimators to perform on-the-fly Doppler broadening in media with spatially continuously variable properties, where the algorithm takes into consideration all of the temperature variance, the coolant density variance and the atomic density variance from such as depletion together. The on-the-fly Doppler broadening runtime data is 3.69 GB, attainable by taking as lower as minutes with the DCT-DB processing code on GPU. Numerical simulations of homogeneous pincell problems, homogeneous 3x3 assemblies mini reactor problems, and pincell problems with variable temperature distributions are validated against the ENDF point-wise cross sections, where the new method costs only 3 to 5 times more computational time than reference problems with uniform temperature valued at the single ENDF point-wise cross section table used, which is competitive with existing methods. The new method's competitiveness over windowed multipole for spatially variable media is justified for on-the-fly Doppler broadening for problems with depleted fuel on GPU devices as well. The widely adopted approach of using average temperatures in fuel, gap, clad and moderator of a pincell could underestimate fission power by 0.04% and k-effective by 45 pcm. In fair comparisons, Apple M1 Pro CPU is 3–6 times more energy efficient than mainstream AMD64 CPUs. The new method is also in favor of current and coming exascale HPC systems with large memory bandwidth. Third, we found a closed form of relationship between multipole parameters and DCT coefficients, which brings a broader picture to existing Doppler broadening methods.