Uncertainty quantification in reactor physics using adjoint/perturbation techniques and adaptive spectral methods

Abstract

This thesis presents the development and the implementation of an uncertainty propagation algorithm based on the concept of spectral expansion. The first part of the thesis is dedicated to the study of uncertainty propagation methodologies and to the analysis of spectral techniques. The concepts introduced within this preliminary analysis are successively used for the derivation of the spectral algorithm. In Chapter 2 we discuss the application of higher order adjoint perturbation theory for coupled problems. This method is relatively easy to implement once the first order adjoint problem is defined, however it is computationally expensive. It is shown, for example, that the number of additional adjoint calculations needed to build the Hessian matrix of a response corresponds, for nonlinear problems, to two times the number of input parameters. It is also shown that for linear problems this number can be halved. It is also discussed that for linear problems it is possible to perform a ranking of the higher order perturbation components, while for nonlinear ones this is not the case. In general, higher order adjoint perturbation theory can be a useful tool to understand uncertainty propagation phenomena. In Chapter 3 an overview of spectral techniques for uncertainty quantification is presented. The mathematical backgrounds of two approaches, defined as intrusive and non-intrusive, are discussed. These approaches are applied to perform uncertainty quantification of a simplified coupled time-dependent problem. The illustrative example shows how non-intrusive approaches are relatively easy to apply while intrusive approaches are quite challenging from the implementation point of view. The curse of dimensionality affecting spectral techniques is also discussed. The example also demonstrates that for time-dependent problems, the convergence of spectral expansions required to represent stochastic outputs varies considerably during the transient. From this point of view, non-intrusive approaches allow the usage of different expansion orders at different times, thereby reducing the computational requirements. Using these initial conclusions as a starting point, an algorithm based on the definition of Polynomial Chaos Expansion is developed. Chapter 4 introduces this new algorithm for the application of quadrature based spectral techniques. This algorithm is based on the notion of sparse grid and its application is divided into two main steps. Firstly, the algorithm adds quadrature points exclusively along the main axes of the stochastic domain. During this phase the convergence of the PCE is assessed and a reduced multi-dimensional PCE is defined. Secondly, this reduced PCE is then used within the second part of the algorithm which focuses on the addition of higher dimensional sub-grids to the final quadrature rule. The adaptive sparse grid algorithm is tested for a reference stochastic case defined by using a simple source detector problem. The algorithm is first validated by comparing it to another sparse grid integration approach found in literature. It is successively shown how the particular construction of the spectral basis, based on a convergence check performed considering each random direction to be independent, can further reduce the number of realizations needed to build the spectral outputs. In Chapter 5 two cost reduction techniques which take advantage of the peculiar definition of the algorithm are presented. These techniques are proven to be effective in the reduction of quadrature points needed to reach convergence. Two uncertainty propagation examples are also considered. The method has been proven to be particularly effective for reactor physics applications, mainly because of the fact that higher order propagation phenomena are usually dominated by a limited set of input parameters. It is also shown, with the first example, that the convergence rate of the adaptive quadrature algorithm directly depends on the differentiability of the response surface. Chapter 6 shows another application of the adaptive sparse grid algorithm, this time to a time-dependent multi-physics problem. This problem is formulated in order to reproduce the type of system that arises when performing safety analysis. Two reference transients simulating an accident scenario of fast reactors are considered. Even in this case the adaptive algorithm proves to be very effective, being capable of reproducing all the stochastic outputs of interest with a relatively low number of realizations. In conclusion, adaptive spectral methods represent a computationally efficient uncertainty quantification technique when in presence of a moderately large set of random input parameters. However, this number strongly depends on the regularity of the response surface. Several strategies could be adopted in order to increase this number and make the method more appealing for a larger set of problems. An overview of these possibilities is presented in the final recommendation section of the thesis.