Implicit stochastic uncertainty propagation scheme for two-step Monte Carlo simulations applied to R2S-UNED

In recent years, two-step Monte Carlo simulations have become increasingly important in nuclear analysis. However, the quantification of stochastic uncertainties in two-step Monte Carlo simulations, in order to evaluate the statistical convergence of final nuclear responses, remains challenging. At present, stochastic uncertainty propagation methods, mainly developed for rigorous-two-step methodologies, exhibit several limitations. In particular, existing methods rely upon the calculation of the covariance matrix of the radiation field estimated in the first Monte Carlo simulation, which is computationally demanding in terms of memory and time. It is important to note that the size of the covariance matrix depends on the discretisation employed to estimate the radiation field and, therefore, computational resources required for calculating the covariance matrix increase with discretization. Indeed, the calculation of the covariance matrix is unfeasible in nuclear analyses for real-world fusion facilities, such as shutdown dose rate calculations using rigorous-two-step methodologies in JET, where large-scale geometries combined with fine discretizations render the size of the covariance matrix impractical. The present paper introduces an innovative methodology, the implicit stochastic uncertainty propagation scheme, to quantify the stochastic uncertainty in the final nuclear response due to the first Monte Carlo simulation, whilst avoiding the calculation of the covariance matrix. The implicit scheme involves the definition of a random variable according to specific criteria. As such, the evaluation of the random variable, as a Monte Carlo tally, allows for quantifying the stochastic uncertainty in the final nuclear response due to the first Monte Carlo simulation. The stochastic uncertainty due to the second Monte Carlo simulation is directly quantified by Monte Carlo codes along with the final nuclear response. Finally, the implicit scheme is implemented in R2S-UNED –a two-step Monte Carlo simulation code designed to calculate the shutdown dose rate resulting from material activation– and, subsequently, employed to analyse the ITER shutdown dose rate benchmark exercise –a well-established test case in which the discretisation of the radiation field renders calculation of the covariance matrix infeasible. The comparison of the ITER benchmark against the brute force method demonstrates the correct performance of the implicit scheme, as well as verifies the applicability in nuclear analyses where existing methods are unfeasible because of the calculation of the covariance matrix.