Alberto Sartori Nuclear Engineering Division, Department of Energy, Politecnico di Milano, Milano 20156, Italy e-mail: alberto.sartori@polimi.it Antonio Cammi Nuclear Engineering Division, Department of Energy, Politecnico di Milano, Milano 20156, Italy e-mail: antonio.cammi@polimi.it Lelio Luzzi 1 Nuclear Engineering Division, Department of Energy, Politecnico di Milano, Milano 20156, Italy e-mail: lelio.luzzi@polimi.it Gianluigi Rozza SISSA MathLab, International School for Advanced Studies, Trieste 34136, Italy e-mail: gianluigi.rozza@sissa.it A Reduced Basis Approach for Modeling the Movement of Nuclear Reactor Control Rods This work presents a reduced order model (ROM) aimed at simulating nuclear reactor control rods movement and featuring fast-running prediction of reactivity and neutron flux distribution as well. In particular, the reduced basis (RB) method (built upon a high-fidelity finite element (FE) approximation) has been employed. The neutronics has been modeled according to a parametrized stationary version of the multigroup neutron diffusion equation, which can be formulated as a generalized eigenvalue problem. Within the RB framework, the centroidal Voronoi tessellation is employed as a sampling technique due to the possibility of a hierarchical parameter space exploration, without relying on a “classical” a posteriori error estimation, and saving an important amount of computa- tional time in the offline phase. Here, the proposed ROM is capable of correctly predicting, with respect to the high-fidelity FE approximation, both the reactivity and neutron flux shape. In this way, a computational speedup of at least three orders of magnitude is achieved. If a higher precision is required, the number of employed basis functions (BFs) must be increased. [DOI: 10.1115/1.4031945] Keywords: reduced basis method, control rod movement, parametrized generalized eigenvalue equation, neutronics, centroidal Voronoi tessellation 1 Introduction In the analysis of nuclear reactor dynamics, the most widespread approach is based on the point-kinetics (PK) equations [1]. This description of the neutronics is based on a set of coupled non- linear ordinary differential equations that describe both the time- dependence of the neutron population in the reactor and the decay of the delayed neutron precursors, allowing for the main feedback reactivity effects. Among the several assumptions entered in the derivation of these equations, the strongest approximation regards the shape of the neutron flux, which is assumed to be represented by a single, time-independent spatial mode. Otherwise, whether the reactors are characterized by complex geometries and asymmetric core configurations, more accurate modeling approaches might be needed to provide more detailed in- sights concerning the reactor behavior during operational transients. It is worth mentioning that the innovative reactor concepts, for instance, Generation IV reactors [2], feature power density and tem- perature ranges, experienced by structural materials, such that the corresponding spatial dependence cannot be neglected. Moreover, in order to develop suitable control strategies for such reactors, the spatial effects induced by the movement of the control rods have to be taken into account as well. Currently, in literature, it is possible to find many attempts in developing nonzero-dimensional reactor models for different appli- cations by means of the modal synthesis method. A comprehensive review of works carried out in the nuclear engineering field can be found in Ref. [3]. The potentials of reduced order techniques, such as proper orthogonal decomposition (POD) [4–6] or RB method [7,8], for developing a spatial kinetics have been addressed in Refs. [9,10]. The present contribution would offer a methodological approach to improve the already-developed simulation tools, which are based on pointwise kinetics, allowing for the spatial effects. The main idea is that the temporal evolution can still be described according to PK equations, but at each time step, the reactivity is estimated by means of a fast-running ROM. Indeed, a ROM for simulating nuclear re- actors control rods movement, which features fast-running compu- tational time for both reactivity and flux shape prediction, has been developed relying on the RB method [7,8]. The reactivity and neu- tron flux shape are computed by solving the stationary version of the multigroup neutron diffusion equation [11], which can be for- mulated as a generalized eigenvalue problem. A two-dimensional (2-D) x-y model featuring two control rods surrounded by fissile material, with reference to a TRIGA Mark II nuclear reactor [12], has been employed. The governing partial differential equa- tions have been parametrized allowing for geometric deformations of the subdomains to model a continuous movement of the control rods, where the heights of the rods are the varying parameters. Within the RB method framework, the centroidal Voronoi tessella- tion has been employed as a sampling technique for the parameter space exploration. All the simulations have been performed relying on a procedure developed on purpose, based on RBniCS [13]. The paper is organized as follows: The RB strategies are first presented in Sec. 2 for a generalized eigenvalue model problem. The parametrized reactor spatial kinetics and the implementation of the RB method are addressed in Sec. 3. The main representative results are given in Sec. 4. Finally, in Sec. 5, the main conclusions and future perspectives are presented. 2 RB Strategies In this section, the fundamental mathematical aspects of the RB method for a generalized eigenvalue problem are addressed. For the sake of brevity, a scalar problem is considered, since the extension to the vectorial case is straightforward. A detailed presentation of 1 Corresponding author. Manuscript received November 24, 2014; final manuscript received October 25, 2015; published online February 29, 2016. Assoc. Editor: Asif Arastu. Journal of Nuclear Engineering and Radiation Science APRIL 2016, Vol. 2 / 021019-1 Copyright © 2016 by ASME Downloaded From: http://nuclearengineering.asmedigitalcollection.asme.org/ on 03/17/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use