Abstract
The diffusion equation turns out to be easier to solve than the Boltzmann equation, especially for large 3D reactors. This explains its widespread use in industrial calculation schemes. We will now deal with a few classical methods—without expatiating on the numerical methods used for solving linear systems (there is abundant literature available on this particular subject matter)—so as to focus on the aspects dealing with neutronics.
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Notes
- 1.
In this chapter, we illustrate the example with the EDF 3D neutronics diffusion code, COCCINELLE .
- 2.
Thomas E. Booth: Power iteration method for the several largest eigenvalues and eigenfunctions, Nuclear Science and Engineering, 154, pp. 48–62 (2006).
- 3.
Mathematically, the Finite Difference method does not require any integration on a control volume but only the substitution of each term in the equation by its discretized form. In practice, this is equivalent to integrating around the position being considered and relates this method to the Finite Volume method.
- 4.
Kord S. Smith : An analytic nodal method for solving the two-group, multidimensional, static and transient neutron diffusion equations, Degree in Nuclear Engineering and Master of Science, MIT (1979). After completing his PhD at MIT (Spatial homogenization methods for light water reactor analysis, 1980), Kord Sterling Smith (1954–), developed several efficient numerical methods for Studsvik Scandpower’s nodal code, SIMULATE . He is a world-renowned expert on nodal methods.
(Courtesy Smith) - 5.
For details, see José Félix Pérez Méndez-Castrillon : Reconstruction of Three-Dimensional flux shapes from nodal solution, Master of Science at the MIT, June 1984.
- 6.
See also Ray G. Gamino : The development and application of Supernodal methods to PWR analysis, PhD at the MIT, May 1986. A good review of models.
- 7.
The transverse integration procedure in Nodal Expansion Method is particularly well described inBernard Ronald Bandini : A Three-Dimensional transient nuetronics routine for the TRAC-PF1 reactor thermalhydauli computer code, PhD at The Pennsylvania State University, 1990, p. 30.
- 8.
A fine summary of nodal methods (including the order-4 method) is set out in Slimane Noceir’s PhD thesis : Sur les méthodes nodales appliquées aux calculs critiques des réacteurs en théorie de la diffusion, [On nodal methods applied to critical reactor calculations in diffusion theory], Thesis, University of Franche-Comté, 1993. After his thesis, S. Noceir was recruited by EDF R&D, where he contributed to the development of the nodal method of order 4 in the COCCINELLE code. He subsequently worked on the thermomechanical code CYRANO3 before contributing to a mixed EDF/CEA project on thermomechanics.
(Courtesy Noceir) - 9.
J.M. Noh, N.Z. Cho : A new approach of analytic basis function expansion to neutron diffusion nodal calculations, Nuclear Science and Engineering, 116, 165 (1994).
- 10.
H.C. Lee , C.H. Kim : Unified nodal method formulation for analytic function expansion nodal method solution to two-group diffusion equations in rectangular geometry, Nuclear Science and Engineering, 140, 137–151 (2002).
- 11.
R.W. Clough: The Finite Element method in plane stress analysis, proceedings of the 2nd ASCE conference on electronic computation, Pittsburgh Sept.8–9, USA, (1960).
- 12.
J. Cartier , G. Samba : Mixed and hybrid finite element method for the transport equation, Nuclear Science and Engineering, 154, pp. 28–47 (2006).
- 13.
V. Kourganoff, Annals of Astrophysics, No. 12, 169 (1949).
- 14.
Vassili S. Vladimirov (1923–), a member of the USSR Academy of Sciences, worked in several domains in applied mathematics, such as linear systems theory and the generalization of the Tauber theorem for multiple dimension functions. In the 1960s, his initial works were on the mathematics of the neutron transport equation, which laid the foundations for the even-odd formulation. Most of his works have been translated in French (Vladimirov 1967, 1979).
V.S. Vladimirov source: Teoreticheskaya i Matematicheskaya Physika 94, 1 (1993), photograph unknown - 15.
Hoan Nguyen-Ngoc : Résolution variationnelle des Equations de diffusion multigroupe indépendantes du temps [Variational resolution of time-independent multi-group diffusion equations], thesis presented to the University of Paris (1965).
- 16.
Christian Labbe: Etude de la mesure d’antiréactivité des barres de la filière à neutrons rapides par la méthode multiplication de source modifiée [Study of the measurement of anti-reactivity of rods in fast-neutron reactor through multiplication of the modified source], PhD thesis, INSTN Grenoble (1979).
- 17.
Han-Sem Joo: Resolution of the control rod cusping problem for nodal methods, PhD at the Massachusetts Institute of Technology (1984) under the direction of Allan F. Henry .
- 18.
Courtesy Marie Hypolite.
- 19.
For the search of the roots of a determinant, the reader is referred to (Traub 1964) or (Durand 1960).
- 20.
Denis Kerdraon (1972–). After a Master’s Degree in Physical Energetics at the Institut National Polytechnique, Grenoble in 1998, he went on to obtain a PhD in the same field at Institut National Polytechnique, Grenoble in 2001 on the physics of hybrid reactors (coupling of a neutron accelerator and a sub-critical reactor) at the Institut des Sciences Nucléaires (IN2P3-ISN), Grenoble. He joined EDF in 2001 and worked on safety studies of the ALCADE and PARITE MOX fuel management systems for the EDF reactor fleet, and on experimental validation of the core calculation chain at EDF/UNIE. In 2006 he joined the R&D department, where he is in charge of development of the GAB application (Automatic Neutron Physics Library Generator) for the CASSIOPEE chain and the future calculation chain, F3C.
- 21.
Jérôme Texeraud (1979–). After completing his studies in mechanics and applied mathematics at the MATMECA engineering school in Bordeaux, he joined EDF/R&D in 2005, where he worked on several models in the COCCINELLE diffusion code, which he managed until 2011.
- 22.
Serge Marguet (1964–). (It is not an easy task to write one’s own bibliographical notes, but let me try anyway!) After his engineering studies in fluid mechanics (ENSHMG, Grenoble 1986), then numerical analysis (ENSIMAG, Grenoble 1987), he was recruited for EDF/DER by Jean-Pierre West in the neutron physics group. He contributed to the development of the 3D diffusion code, COCCINELLE, and to the calculation chain over a number of years. After a second period at CEA/Cadarache working on the fuel cycle code DARWIN, he returned to Clamart, where he designed the STRAPONTIN residual power code. He was then promoted head of the Severe Accidents project from 2000 to 2003, and is a designated European expert in this field. In 2007, he resumed his work the COCCINELLE code, promoting the integration of parallelism in 2012 for simulators and an online piloting tool. He teaches neutron physics at the Ecole Nationale Supérieure de Risques Industriels in Bourges and is the author of two works in the nuclear field: the present book on reactor physics of a popularizing work on severe accidents.
(Courtesy Smith)
(Courtesy Noceir)
V.S. Vladimirov source: Teoreticheskaya i Matematicheskaya Physika 94, 1 (1993), photograph unknownBibliography
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Marguet, S. (2017). Computation Methods in Diffusion Theory. In: The Physics of Nuclear Reactors. Springer, Cham. https://doi.org/10.1007/978-3-319-59560-3_18
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