Homogenized Diffusion Approximations to the Neutron Transport Equation

Larsen, Edward W.; Hughes, R. P.

doi:10.13182/nse80-a19851

Cited by 16 publications

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“…Since the the homogenized problem has constant coe cients, we know its exact rst eigenvalue and eigenvector (a sine function). Then we re-construct the uxes by the following homogenized approximation The constants D and are given by formulas (21) and (20), whereas corrector is given by equation (22).…”

Section: Numerical Resultsmentioning

confidence: 99%

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Homogenization of a spectral problem in neutronic multigroup diffusion

Allaire

Capdeboscq

2000

Computer Methods in Applied Mechanics and Engineering

127

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This paper is concerned with the homogenization of an eigenvalue problem in a periodic heterogeneous domain for the multigroup neutron di usion system. Such a model is used for studying the criticality of nuclear reactor cores. We prove that the rst eigenvector of the multigroup system in the periodicity cell controls the oscillatory behavior of the solutions, whereas the global trend is asymptotically given by a homogenized di usion eigenvalue problem. The neutron ux, corresponding to the rst eigenvector of the multigroup system, tends to the product of the rst periodic and homogenized eigenvectors. This result justi es and improves the engineering procedure used in practice for nuclear reactor core computation.

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Section: Numerical Resultsmentioning

confidence: 99%

“…In Theorem 3.2 they are de ned as the solutions of system ( 22), whereas in Proposition 5.3 they are solutions of system (52). Our notations are consistent in the sense that (52) is just (22), each line being multiplied by .…”

Section: Homogenizationmentioning

confidence: 99%

Homogenization of a spectral problem in neutronic multigroup diffusion

Allaire

Capdeboscq

2000

Computer Methods in Applied Mechanics and Engineering

127

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“…Also, in the work by Larsen and others already investigated the asymptotic regime of kinetic equation leading to diffusion equations. 23,24,17 In particular, it has already been pointed out that the transport coefficients only depend on mean and standard deviations of the underlying probability distributions, here, on the ones of T and W, respectively. This has also been analyzed for numerical schemes for example in Refs.…”

Section: Remark 32mentioning

confidence: 99%

Large time behavior of averaged kinetic models on networks

Herty

Ringhofer

2015

Math. Models Methods Appl. Sci.

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Communicated by P. DegondWe are interested in flows on general networks and derive a kinetic equation describing general production, social or transportation networks. Corresponding macroscopic transport equations for large time and homogenized behavior are obtained and studied numerically. This work continues a recent discussion [Averaged kinetic models for flows on unstructured networks, Kinetic Related Models 4 (2011) 1081-1096] and provides additionally explicit equilibrium solutions, second-order macroscopic approximations as well as numerical simulations on a large scale homogenized network.Possible applications are production networks. Here, a good is flowing from a raw material supplier through a certain number of layers (nodes in the networks) of intermediate producers to a final consumer. Due to possible machine breakdowns and service distributions travel and waiting times of goods may be described using probability distributions. Production networks of this type have been introduced originally in Refs. 5,8, 9 and 11, and optimized in Refs. 15 and 16. Another application might be air traffic where we consider passengers arriving and leaving airports. The links are the possible flight routes and waiting times reflect delays at airports due to, for example, stochastic weather conditions. A description of effects of possible network structures in application examples for traffic and social networks has been studied in Ref. 7. Therein, the so-called small world network structure is discussed. Those networks are constructed by a process described in Ref. 4 and the mean connection between nodes grows at most logarithmic with the number of nodes. 30 Within this paper we want to explore the connection between a simple dynamics with possible applications in production and air traffic and the structure of the network more closely. To this end we consider a general mathematical model for transport on graphs described as a multi-agent model. We apply asymptotic techniques (borrowed from classical kinetic theory) to derive a simplified model for flows on large networks on large time scales. This reduces the computational complexity of the study of long time phenomena in such flows. We develop a kinetic model for flows on arbitrary graphs, originally proposed in Ref. 20, under some simplifying assumptions, which make the large time scale model practically applicable. The macroscopic model consists of a convection-diffusion equation for the agent density, posed on a continuum in space, representing the graph of the network. In order for this model to be "reasonably smooth", i.e. not to involve measure-valued transport coefficients, it is necessary to locate the nodes of the graph in a certain way, i.e. to "draw the graph in R 2 " in a special way. Reorganizing the network this way results in the solution of an optimization problem for the coordinates of the network nodes. This represents the generalization of an idea, originally proposed in Ref. 7 to higher dimensions. The final macroscopic model is a convection-d...

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“…In many problems, it is necessary an homogenization process which consist in approximating the heterogeneous real problem by a fictitious homogeneous problem. To do that, it is required to define homogenized cross sections and homogenized diffusion coefficients, and the optimal definition of these parameters is very discussed due to, even though different definitions work well in some particular cases [Larsen and Hughes, 1980], there is no one definition valid for any application. In any case, this discussion is beyond the scope of this thesis.…”

Section: Introduction To the Neutron Diffusion Equationmentioning

confidence: 99%

Contributions to solve the Multi-group Neutron Transport equation with different Angular Approaches

Rafet¹

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The most accurate way to know the movement of the neutrons through matter is achieved by solving the Neutron Transport Equation. Three different approaches to solve this equation have been investigated in this thesis: Discrete Ordinates Neutron Transport Equation, Neutron Diffusion Equation and Simplified Spherical Harmonics Equations.In order to solve the equations, different schemes of the Finite Differences Method were studied. The solution of these equations describes the population of neutrons and the occurred reactions inside a nuclear system. These variables are related with the flux and power, fundamental parameters for the Nuclear Safety Analysis.Por último, pero no menos importante, agradezco el apoyo de mi familia, mis hermanos y mi padre; y a los que me hacen sentir como tal, Jesús y Nati. Muy especialmente quiero dar las gracias a María, por estar siempre a mi lado e invitar a superarme. Finalmente, sería injusto no acordarme de mi madre a quien debo todo lo que soy.

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Homogenized Diffusion Approximations to the Neutron Transport Equation

Cited by 16 publications

References 0 publications

Homogenization of a spectral problem in neutronic multigroup diffusion

Homogenization of a spectral problem in neutronic multigroup diffusion

Large time behavior of averaged kinetic models on networks

Contributions to solve the Multi-group Neutron Transport equation with different Angular Approaches

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