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Reduced Basis Approaches in Time-Dependent Non-Coercive Settings for Modelling the Movement of Nuclear Reactor Control Rods

Abstract

In this work, two approaches, based on the certified Reduced Basis method, have been developed for simulating the movement of nuclear reactor control rods, in time-dependent non-coercive settings featuring a 3D geometrical framework. In particular, in a first approach, a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod. In the second approach, a “staircase” strategy has been adopted for simulating the movement of all the three rods featured by the nuclear reactor chosen as case study. The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion, which, in the present case, is a set of ten coupled parametrized parabolic equations (two energy groups for the neutron flux, and eight for the precursors). Both the reduced order models, developed according to the two approaches, provided a very good accuracy compared with high-fidelity results, assumed as “truth” solutions. At the same time, the computational speed-up in the Online phase, with respect to the fine “truth” finite element discretization, achievable by both the proposed approaches is at least of three orders of magnitude, allowing a real-time simulation of the rod movement and control.

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Corresponding author

*Corresponding author. Email addresses: alberto.sartori@polimi.it (A. Sartori), antonio.cammi@polimi.it (A. Cammi), lelio.luzzi@polimi.it (L. Luzzi), gianluigi.rozza@sissa.it (G. Rozza)

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[1] Barrault, M., Maday, Y., Nguyen, N. C., and Patera, A. T.. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique, 339(9):667672, 2004.
[2] Bell, G. and Glasstone, S.. Nuclear reactor theory. Van Nostrand Reinhold Co., 1970.
[3] Chatterjee, A.. An introduction to the proper orthogonal decomposition. Current Science, 78:808817, 2000.
[4] Dautray, R. and Lions, J. L.. Mathematical Analysis and Numerical Methods for Science and Technology: Volume 6 Evolution Problems II. Springer-Verlag Berlin Heidelberg, 2000.
[5] Duderstadt, J. J. and Hamilton, L. J.. Nuclear Reactor Analysis. John Wiley and Sons, New York, 1976.
[6] Gelsomino, F. and Rozza, G.. Comparison and combination of reduced-order modelling techniques in 3D parametrized heat transfer problems. Mathematical and Computer Modelling of Dynamical Systems, 17(4):371394, 2011.
[7] General Atomic Company, U.S.A. TRIGA Mark II Reactor General Specifications and Description, 1964.
[8] Geuzaine, C. and Remacle, J. F.. Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities. International Journal for Numerical Methods in Engineering, 79(11):13091331, 2009.
[9] GIF. A Technology Roadmap for Generation IV Nuclear Energy System. Technical report, GIF-002-00, 2002.
[10] Grepl, M. A. and Patera, A. T.. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis, 39:157181, 1 2005.
[11] Haasdonk, B. and Ohlberger, M.. Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Mathematical Modelling and Numerical Analysis, 42(2):277302, 2008.
[12] Holmes, P., Lumley, J., and Berkooz, G.. Turbolence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, 1996.
[13] Huynh, D., Knezevic, D., Chen, Y., Hesthaven, J. S., and Patera, A.. A natural-norm successive constraint method for inf-sup lower bounds. Computer Methods in Applied Mechanics and Engineering, 199(29):19631975, 2010.
[14] Huynh, D. B. P., Rozza, G., Sen, S., and Patera, A. T.. A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants. Comptes Rendus Mathematique, 345(8):473478, 2007.
[15] Kirk, B. S., Peterson, J. W., Stogner, R. H., and Carey, G. F.. libMesh: A C++ Library for Parallel Adaptive Mesh Refinement/Coarsening Simulations. Engineering with Computers, 22(3–4):237254, 2006.
[16] Knezevic, D. J. and Peterson, J. W.. A high-performance parallel implementation of the certified reduced basis method. Computer Methods in Applied Mechanics and Engineering, 200:14551466, 2011.
[17] Koning, A., Forrest, R., Kellett, M., Mills, R., Henriksson, H., and Rugama, Y.. The JEFF-3.1 Nuclear Data Library. Technical Report NEA – OECD, JEFF Report 21, 2006.
[18] Krane, K. S.. Introductory nuclear physics. Wiley, India, 1987.
[19] Lamarsh, J. R.. Introduction to nuclear reactor theory, 3rd Edition. Addison-Wesley Publishing Company, 1977.
[20] Lamarshi, J. R. and Baratta, A. J.. Introduction to nuclear engineering. Prentice-Hall, New Jersey, 2001.
[21] Lassila, T., Manzoni, A., and Rozza, G.. On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition. ESAIM: Mathematical Modelling and Numerical Analysis, 46(6):15551576, 2012.
[22] Maday, Y.. Reduced basis method for the rapid and reliable solution of partial differential equations. In Proceedings of International Conference of Mathematicians, Madrid. European Mathematical Society Eds., 2006.
[23] Manzoni, A., Quarteroni, A., and Rozza, G.. Computational Reduction for Parametrized PDEs: Strategies and Applications. Milan Journal of Mathematics, 80(2):283309, 2012.
[24] Nguyen, N. C., Rozza, G., Huynh, D. B. P., and Patera, A. T.. Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real-Time Bayesan Parameter Estimation, John Wiley & Sons, Ltd, 2010, Ch. 8, 157185.
[25] Nguyen, N. C., Rozza, G., and Patera, A. T.. Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation. Calcolo, 46(3):157185, 2009.
[26] Patera, A. T. and Rozza, G.. Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations, MIT Pappalardo Graduate Monographs in Mechanical Engineering, online at http://augustine.mit.edu, 2007.
[27] Quarteroni, A., Rozza, G., and Manzoni, A.. Certified reduced basis approximation for parametrized partial differential equations and applications. Journal of Mathematics in Industry, 1(1):149, 2011.
[28] Quarteroni, A. and Valli, A.. Numerical approximation of partial differential equations. Volume 23. Springer, 2008.
[29] Rozza, G.. Fundamentals of Reduced Basis Method for problems governed by parametrized PDEs and applications. In CISM Lectures notes “Separated Representation and PGD based model reduction: fundamentals and applications”. Chinesta, F. and Ladeveze, P. (eds.), Springer Vienna, 2014.
[30] Rozza, G., Huynh, D., and Patera, A.. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archives of Computational Methods in Engineering, 15(3):147, 2008.
[31] Sartori, A., Baroli, D., Cammi, A., Chiesa, D., Luzzi, L., Ponciroli, R., Previtali, E., Ricotti, M. E., Rozza, G., and Sisti, M.. Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics. Annals of Nuclear Energy, 71:217229, 2014.
[32] Sartori, A., Baroli, D., Cammi, A., Luzzi, L., and Rozza, G.. A Reduced Order Model for Multi-Group Time-Dependent Parametrized Reactor Spatial Kinetics. In Proceedings of the 2014 22nd International Conference on Nuclear Engineering (ICONE22), Prague, Czech Republic, July 7-11, 2014, Paper 30707, ©ASME 2014.
[33] Sartori, A., Cammi, A., Luzzi, L., and Rozza, G.. Multi-physics reduced order models for analysis of Lead Fast Reactor single channel. Annals of Nuclear Energy, 87:198208, 2016.
[34] Schultz, M. A.. Nuclear reactor kinetics and control. McGraw-Hill, 1961.
[35] PSG2 / Serpent Monte Carlo Reactor Physics Burnup Calculation Code, 2011. URL http://montecarlo.vtt.fi.

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Reduced Basis Approaches in Time-Dependent Non-Coercive Settings for Modelling the Movement of Nuclear Reactor Control Rods

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