Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T02:11:14.308Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical methods in calculus of variations and optimal control Computer aspects of numerical algorithms

Published online by Cambridge University Press:  22 June 2016

Alberto Sartori ,
Antonio Cammi ,
Lelio Luzzi  and
Gianluigi Rozza
Alberto Sartori*
Affiliation:
Politecnico di Milano – Department of Energy, CeSNEF, Enrico Fermi Center for Nuclear Studies, via La Masa 34, 20156 Milano, Italy SISSA, International School for Advanced Studies, Mathematics Area – mathLab, Via Bonomea 265, 34136 Trieste, Italy
Antonio Cammi*
Affiliation:
Politecnico di Milano – Department of Energy, CeSNEF, Enrico Fermi Center for Nuclear Studies, via La Masa 34, 20156 Milano, Italy
Lelio Luzzi*
Affiliation:
Politecnico di Milano – Department of Energy, CeSNEF, Enrico Fermi Center for Nuclear Studies, via La Masa 34, 20156 Milano, Italy
Gianluigi Rozza*
Affiliation:
SISSA, International School for Advanced Studies, Mathematics Area – mathLab, Via Bonomea 265, 34136 Trieste, Italy
*
*Corresponding author. Email addresses:[email protected] (A. Sartori), [email protected] (A. Cammi), [email protected] (L. Luzzi), [email protected] (G. Rozza)
*Corresponding author. Email addresses:[email protected] (A. Sartori), [email protected] (A. Cammi), [email protected] (L. Luzzi), [email protected] (G. Rozza)
*Corresponding author. Email addresses:[email protected] (A. Sartori), [email protected] (A. Cammi), [email protected] (L. Luzzi), [email protected] (G. Rozza)
*Corresponding author. Email addresses:[email protected] (A. Sartori), [email protected] (A. Cammi), [email protected] (L. Luzzi), [email protected] (G. Rozza)
Get access

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.

Keywords

Reduced basis methodcontrol rod movementspatial kineticsparametrized geometrymulti-group neutron diffusionnon-coercive operators

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65Y20: Complexity and performance of numerical algorithms 49M25: Discrete approximations
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 1 , July 2016 , pp. 23 - 59
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[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):667–672, 2004.CrossRefGoogle Scholar
[2] Bell, G. and Glasstone, S.. Nuclear reactor theory. Van Nostrand Reinhold Co., 1970.Google Scholar
[3] Chatterjee, A.. An introduction to the proper orthogonal decomposition. Current Science, 78:808–817, 2000.Google Scholar
[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.Google Scholar
[5] Duderstadt, J. J. and Hamilton, L. J.. Nuclear Reactor Analysis. John Wiley and Sons, New York, 1976.Google Scholar
[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):371–394, 2011.Google Scholar
[7] General Atomic Company, U.S.A. TRIGA Mark II Reactor General Specifications and Description, 1964.Google Scholar
[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):1309–1331, 2009.Google Scholar
[9] GIF. A Technology Roadmap for Generation IV Nuclear Energy System. Technical report, GIF-002-00, 2002.Google Scholar
[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, 39:157–181, 1 2005.Google Scholar
[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):277–302, 2008.Google Scholar
[12] Holmes, P., Lumley, J., and Berkooz, G.. Turbolence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, 1996.CrossRefGoogle Scholar
[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):1963–1975, 2010.Google Scholar
[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):473–478, 2007.Google Scholar
[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):237–254, 2006.Google Scholar
[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, 200:1455–1466, 2011.Google Scholar
[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.Google Scholar
[18] Krane, K. S.. Introductory nuclear physics. Wiley, India, 1987.Google Scholar
[19] Lamarsh, J. R.. Introduction to nuclear reactor theory, 3rd Edition. Addison-Wesley Publishing Company, 1977.Google Scholar
[20] Lamarshi, J. R. and Baratta, A. J.. Introduction to nuclear engineering. Prentice-Hall, New Jersey, 2001.Google Scholar
[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):1555–1576, 2012.Google Scholar
[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.Google Scholar
[23] Manzoni, A., Quarteroni, A., and Rozza, G.. Computational Reduction for Parametrized PDEs: Strategies and Applications. Milan Journal of Mathematics, 80(2):283–309, 2012.Google Scholar
[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, 157–185.Google Scholar
[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):157–185, 2009.Google Scholar
[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.Google Scholar
[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):1–49, 2011.CrossRefGoogle Scholar
[28] Quarteroni, A. and Valli, A.. Numerical approximation of partial differential equations. Volume 23. Springer, 2008.Google Scholar
[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.Google Scholar
[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):1–47, 2008.Google Scholar
[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, 71:217–229, 2014.Google Scholar
[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.Google Scholar
[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, 87:198–208, 2016.Google Scholar
[34] Schultz, M. A.. Nuclear reactor kinetics and control. McGraw-Hill, 1961.Google Scholar
[35] PSG2 / Serpent Monte Carlo Reactor Physics Burnup Calculation Code, 2011. URL http://montecarlo.vtt.fi.Google Scholar