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This paper aims at extending the work performed by means of the A N Boundary Element – Response Matrix method in a previous paper, devoted to criticality problems, to non-homogeneous problems ( i.e. the problems that involve a fixed external source, as in case of the accelerator driven systems). As in the preceding paper, the interest is given to the 3D, xyz multigroup systems in which linearly anisotropic scattering is allowed. Standard interface conditions have been adopted and no external intervention in order to improve the numerical results, such as some kind of discontinuity factors, has been introduced. The latter choice is motivated by the need of establishing a clean test of the performances of the A N method in itself. Results of 2D and 3D multigroup fixed source problems obtained with the method described in this paper are compared with those obtained by well assessed reference codes such as the MCNP Monte Carlo code and the PARTISN discrete ordinates code. Even in the absence of discontinuity factors and despite of the intrinsically approximate character of the A N method, the accuracy of the numerical results is more than satisfactory. Finally the computational time is in most cases much shorter than that required by the chosen reference code.

A method for Monte Carlo sensitivity analyses of α-eigenvalue (prompt neutron time decay constant) in a subcritical system is developed using the first-order differential operator sampling (DOS) method. The first-order derivative of α-eigenvalue with respect to nuclear data is calculated using the DOS method that includes the capability of calculating perturbed source effect. This paper is an extension of the author’s previous work for development of the sensitivity analysis method for keff-eigenvalue. Unlike the conventional Monte Carlo method for α-eigenvalue calculation that uses the power iteration of fission sources, this paper introduces a recently developed “time source method”. The “time source method” has a weakness for a void-containing subcritical system, which is overcome by assigning a virtual total cross section in the void region. The perturbed source effect, which is caused by the change of nuclear data in a subcritical system, can be calculated by two methods, the source perturbation iteration method and the superhistory method. The source perturbation iteration method is superior in terms of computation efficiency, but a huge computer memory is required. The superhistory method dramatically reduces the memory requirement, although it worsens the variance of the sensitivity coefficients. The method developed in this paper is applied to some numerical tests that use multi-group constants, and it is verified by comparing to the results obtained by a deterministic perturbation theory.

Several works have been done for the development of models that generalize the Maxwell-Boltzmann distribution, aimed at encompassing physical phenomena that lie outside the thermal equilibrium. Amongst these, there are distributions that result from the non-extensive statistics of Tsallis and Kaniadakis. Starting from these generalized distributions, a Doppler broadening function was proposed in recent papers, using the deformed Kaniadakis distribution, which was numerical, evaluated with the use of a Gauss-Legendre quadrature. From this perspective, this paper presents an analytical solution for the generalized Doppler broadening function through obtaining a partial differential equation, considering the Kaniadakis distribution. This equation is solved analytically using the methods of Frobenius and variation of parameters, in order to obtain a generalized solution for the Doppler broadening function, containing a deformation parameter κ , that measures the deviation in relation to the Maxwell-Boltzmann distribution. Finally, the results were produced considering several values for κ , with the intent of making a comparison with the reference values. For the validation of the deformed Doppler broadening function’s analytical solution, a numerical solution of the partial differential equation was generated. It was possible to use this numerical solution as a benchmark for the analytical solution that was derived. It was demonstrated that the analytical solution obtained is consistent, because when κ tends to zero, the solution falls in the conventional form, when the Maxwell-Boltzmann distribution is considered. Apart from this, the results were shown to be good, especially when we consider the temperature and power ranges for practical applications, as the maximum error obtained was smaller than 1%.

Bateman linear chain solution is widely used for fusion activation calculations. Its first order derivatives with respect to the decay and cross section constants are implemented for the sensitivity analysis. The general formula for these derivatives is a recursive one which is lengthy and cumbersome to use. The large numbers of arithmetic operations consume considerable computational effort and grow as k 2 , where ‘k’ is the size of a linear chain. However, they provide a complete analytical treatment of the sensitivity analysis. In the present paper, the general formula for the derivatives has been reworked and further simplified without any approximations. The resulting derivatives are a set of simple yet exact recursive relations that require almost no computational labour and grow almost linearly with size ‘k’ of the chain. The derived relations can be easily used for a quicker sensitivity analysis and results are presented within the paper. Higher order derivatives can be derived using simple algebra from the first order derivatives. Further, it is shown in the paper that the first and second order derivatives of the Bateman solution can be used within a Taylor series expansion to perform multi-point analysis on fine meshes over a large geometry. Most of the current multi-point algorithms focus on the coupling or algorithmic aspects to accelerate the calculations. In the present scheme, we use taylor expansion of the Bateman solution itself to directly affect the increase in computational speed. The solution of the Bateman equations at a given reference point within the geometry can be used to estimate the results at neighbouring points through a Taylor expansion. While the method in itself is straightforward, it is made feasible with the simplified derivative relations. This methodology has been successfully implemented within the linear chain solution algorithm and a few results and inferences are provided along with the computational accuracy and efficiency.

In this paper, a discontinuous Galerkin finite element method (DGFEM) based on B-spline wavelet on the interval (BSWI) is developed and applied to solve the first-order neutron transport equation with discrete ordinate (SN) angular discretization. Then, using the scaling function of BSWI function as the basis function instead of the usual Lagrange basis function, the neutron transport equation based on BSWI discontinuous finite element is derived. Owing to the high precision characteristic of BSWI function, the method proposed in this paper can obtain fast convergence and statistical numerical accuracy under the condition that few spatial elements are used, and comparing with the results of the traditional discontinuous finite element method. Finally, some numerical examples are utilized to verify the accuracy and efficiency of the proposed method, and the results show that the method has high efficiency, reliable and stability, and the calculation accuracy is higher than that of the conventional discontinuous finite element method.

The Variational Nodal Method (VNM) is generalized for heterogeneous nodes and applied to four kinds of problems including Molten Salt Reactor (MSR) core problem with continuous cross section profile, Pressurized Water Reactor (PWR) control rod cusping effect problem, PWR whole-core pin-by-pin problem, and heterogeneous PWR core problem without fuel-coolant homogenization in each pin cell. Two approaches have been investigated for the treatment of the nodal heterogeneity in this paper. To concentrate on spatial heterogeneity, diffusion approximation was adopted for the angular variable in neutron transport equation. To provide demonstrative numerical results, the codes in this paper were developed in slab geometry. The first method, named as function expansion (FE) method, expands nodal flux by orthogonal polynomials and the nodal cross sections are also expressed as spatial depended functions. The second path, named as finite sub-element (FS) method, takes advantage of the finite-element method by dividing each node into numbers of homogeneous sub-elements and expanding nodal flux into the combination of linear sub-element trial functions. Numerical tests have been carried out to evaluate the ability of the two nodal (coarse-mesh) heterogeneous VNMs by comparing with the fine-mesh homogeneous VNM. It has been demonstrated that both heterogeneous approaches can handle heterogeneous nodes. The FE method is good at continuous-changing heterogeneity as in the MSR core problem, while the FS method is good at discontinuous-changing heterogeneity such as the PWR pin-by-pin problem and heterogeneous PWR core problem. For problems with only one or two discontinuity points such as the PWR control rod cusping effect problem, both of the two methods can catch the effect with high efficiency.

Dependence assessment among human errors plays an important role in human reliability analysis. When dependence between two sequent tasks exists in human reliability analysis, if the preceding task fails, the failure probability of the following task is higher than success. Typically, three major factors are considered: “Closeness in Time” (CT), “Task Relatedness” (TR) and “Similarity of Performers” (SP). Assume TR is not changed, both SP and CT influence the degree of dependence level and SP is discounted by the time as the result of combine two factors in this paper. In this paper, a new computational model is proposed based on the Dempster–Shafer Evidence Theory (DSET) and Evidence Credibility Decay Model (ECDM) to assess the dependence between tasks in human reliability analysis. First, the influenced factors among human tasks are identified and the basic belief assignments (BBAs) of each factor are constructed based on expert evaluation. Then, the BBA of SP is discounted as the result of combining two factors and reconstructed by using the ECDM, the factors are integrated into a fused BBA. Finally, the dependence level is calculated based on fused BBA. Experimental results demonstrate that the proposed model not only quantitatively describe the fact that the input factors influence the dependence level, but also exactly show how the dependence level regular changes with different situations of input factors.

The aim of this paper is to present the critical verification for TRX and BAPL benchmarks between the ENDF/B-VIII.0 and ENDF/B-VII.0 library. Using DRAGON5.0.6 code with 281-group WIMS-D and DRAG library, the integral parameters for TRX and BAPL benchmarks are compared to the Monte-Carlo code and experimental values. For the verification of DRAGON code, the influence of tracking method, resonance self-shielding method, and leakage model are investigated in this paper. The results indicated that the combination methods of collision probability method, equivalence theory, and P1 leakage model have better accuracy in TRX and BAPL benchmarks. For the comparison of ENDF/B-VII.0 and ENDF/B-VIII.0 library, the difference of integral parameters is small between the ENDF/B-VII.0 and ENDF/B-VIII.0 library. The relative error of ρ28 and δ25 is less than 1%. However, the accuracy of δ25 and C* is underestimated by about 1.0%∼1.5% in ENDF/B-VIII.0 library. It means that the fission cross section for U-235 is overestimated in ENDF/B-VIII.0 library and the 2-group cross section comparison is consistent with this tendency. Therefore, the absorption cross section for whole region is increased in thermal energy group, which finally leads to the keff and kinf decreased in ENDF/B-VIII.0 library. Compared with the experimental values in TRX and BAPL cases, the integral parameters in ENDF/B-VIII.0 are more accurate than the ENDF/B-VII.0 library.

The relation between the Rossi alpha and Feynman alpha formulae is well known. The second factorial moment of counts in an interval, which is related to the variance, can be obtained by a double integration of the Rossi alpha expression. However in a recent paper, Munoz Cobo et al. have cast doubts on the generality of this relation stating that this relation is valid only for point models and is not true for space dependent models. The present paper revisits the issue. We examine the multi mode formulae obtained by Munoz Cobo et al. in order to assess the validity of the above relationship. We show that indeed it is possible to derive the Feynman alpha formula by a double integration of the Rossi alpha formula followed by some elementary algebraic manipulation. To demonstrate the generality of this relationship we use two distinct approaches. In the first of these, we show, using the (backward) stochastic transport equation and the space dependent Bartlett formula, that the Rossi alpha and Feynman alpha functions obey the above relationship. In the second, we start from the theory of stochastic point processes and show that the relationship is quite general.

In this paper, a qualitative approach has been used to explore nuclear reactor behaviors with nonlinear feedback. First, a system of four dimensional ordinary differential equations governing the dynamics of a typical nuclear reactor is introduced. These four state variables are the relative power of the reactor, the relative concentration of delayed neutron precursors, the fuel temperature and the coolant temperature. Then, the qualitative behaviors of the dynamical system near its equilibria have been studied analytically by using local bifurcation theory and Sotomayor’s theorem. The results indicated that despite the uncertainty of the reactivity, we can analyze the qualitative behavior changes of the reactor from the bifurcation point of view. Notably, local bifurcations that were considered in this paper include transcritical bifurcation, saddle-node bifurcation and pitchfork bifurcation. The theoretical analysis showed that these three types of local bifurcations may occur in the four dimensional dynamical system. In addition, to confirm the analytical results the numerical simulations are given.

This paper provides an analysis of the generation-to-generation correlations as observed when solving full core eigenvalue problems on PWR systems. Many studies have in the past looked at the impact of these correlations on reported variance and this paper extends the analysis to the observed convergence rate on the tallies, the effect of tally size and the effect of generation size. Since performing meaningful analysis on such a large problem is inherently difficult, a simple homogeneous reflective cube problem with analytical solution was developed that exhibits similar behavior to the full core PWR benchmark. The data in this problem was selected to match the dimensionality of the reactor problem and preserve the migration length travelled by neutrons. Results demonstrate that the variance will deviate significantly from the 1/N (N being the number of simulated particles) convergence rate associated with truly independent generations, but will eventually asymptote to 1/N after 1000's of generations regardless of the numbers of neutrons per generation. This indicates that optimal run strategies should emphasize lower number of active generations with greater number of neutrons per generation to produce the most accurate tally results. This paper also describes and compares three techniques to evaluate suitable confidence intervals in the presence of correlations, one based on using history statistics, one using generation statistics and one batching generations to reduce batch-to-batch correlation. Keywords: Monte Carlo, Tally Convergence, Autocorrelation, Confidence Intervals, United States. Department of Energy (Consortium for Advanced Simulation of Light Water Reactors. Contract DE-AC05-00OR22725)

A general nodal expansion method (GNEM) is developed to solve the multi-dimensional steady and transient convection–diffusion equation in this paper. The developed GNEM is an integration of the modified nodal integral method (MNIM) and nodal expansion method (NEM). Firstly, the local analytical solution of the transverse integral equation from MNIM is obtained under the framework of the basis function expansion method in NEM. Secondly, as to GNEM for time-dependent problems, the full space–time nodal method is adopted. Thirdly, GNEM borrows the nodal balance equation from NEM to ensure that the uniqueness of nodal-average variables and the conservation of all the pseudo-sources in each direction are automatically satisfied, which saves the GNEM from the complex derivation and calculations of the nodal-average variables and the zeroth order pseudo-source terms in MNIM. In this paper, GNEM has integrated the multi-dimensional transient and steady problems into the unified discrete formulation. Besides, the code based on GNEM can be easily developed from the existing NEM code since GNEM-base models have a unified formulation with conventional NEM—they only differ in the two control parameters. Meanwhile, this kind of unified formulation also makes it easier to select GNEM and conventional NEM in different computational domains for a test problem. Finally GNEM is formally developed and numerical results show that GNEM has an accuracy consistent with the MNIM, or even higher accuracy than MNIM.

Shark-X is a set of Perl-based tools build around the deterministic neutron transport code CASMO-5, used to perform uncertainty quantification for lattice calculations at the Paul Scherrer Institut (PSI). While perturbing an input parameter, e.g. a cross section in a given energy range, the resulting perturbation to the effective cross section, e.g. multigroup cross section, can be expressed as the sum of an explicit and implicit component. The implicit term stems from changes in the neutron flux resulting from the cross section perturbation. This effect is neglected when using the standard perturbation theory methodology, even though its importance for uncertainty quantification is reported in the literature. The objectives of the paper are, in view of further enhancing Shark-X, to review a set of existing methods and to propose novel methodologies in order to accurately take the implicit effect into account in the resonance shielding treatment of CASMO-5; as well as to assess the importance of the new tool for typical LWR applications. Amongst the methods considered, those based on an equivalence in dilution allow perturbing the group-averaged cross sections without having to solve the slowing down problem in conjunction with the perturbed point-wise cross sections. However, such methods are found to show only limited improvements as compared to the standard perturbed self-shielded cross sections determined neglecting the implicit effect. As a result, a simpler approach based on the pre-determination of perturbation factors is adopted and implemented into Shark-X and CASMO-5. The new implementation is successfully tested and verified by comparing k-inf sensitivity coefficients computed with Shark-X with those from a Monte-Carlo reference solution, and this for a range of simple homogeneous mixtures as well as for LWR pin cells and lattices. Finally, the paper demonstrates through simple considerations, that the magnitude of the implicit effect is related to the energy group structure in the resolved resonance region, which might be specific to the code used to compute the sensitivity coefficients. The finer is the energy mesh; the smaller is also the impact of the implicit effect, and as such the necessity for a specific treatment.

This paper presents a novel approach devised to solve the transport of neutral particles in scattering and absorbing media. The solution to the linear Boltzmann equation is sought starting from a multi-collision approach of the integro-differential equation which is combined with an approximate model for the description of the residue after truncation of the Neumann series. In the paper, the theoretical basis of such hybrid method is discussed together with the physical intuition at the basis of the methodology. Results for both steady-state and transient problems are presented and an extension to general multi-dimensional, anisotropic problem is reported.

A new Monte Carlo method to calculate two types of kinetics parameters, the effective delayed neutron fraction (βeff) and the prompt neutron generation time (Λ), is proposed in this paper. The new method uses perturbation techniques in which extra delayed neutrons or a fictitious 1/v-absorber is added to the unperturbed system to calculate the βeff or Λ, respectively. In the new method, the perturbation is added as a complex-valued perturbation. This paper conjectures that the change in the eigenvalue due to the perturbation is accurately approximated by the imaginary part of the eigenvalue of the complex-valued perturbed equation. The conjecture is corroborated by certain numerical tests presented in this paper. A Monte Carlo calculation algorithm is established to solve the complex-valued perturbed eigenvalue equation. One single Monte Carlo calculation to solve the complex-valued eigenvalue equation yields highly accurate approximations of the exact kinetics parameters with much less computational costs compared with the previously proposed method that uses multiple Monte Carlo runs.

This paper emphasizes on finding the solution for a stationary transport equation using the technique of Haar wavelet Collocation Method (HWCM). Haar wavelet Collocation Method is efficient and powerful in solving wide class of linear and nonlinear differential equations. Recently Haar wavelet transform has gained the reputation of being a very effective tool for many practical applications. This paper intends to provide the great utility of Haar wavelets to nuclear science problem. In the present paper, two-dimensional Haar wavelets are applied for solution of the stationary Neutron Transport Equation in homogeneous isotropic medium. The proposed method is mathematically very simple, easy and fast. To demonstrate about the efficiency of the method, one test problem is discussed. It can be observed from the computational simulation that the numerical approximate solution is much closer to the exact solution.

In the analysis of the nuclear safety of complex nuclear systems, almost one-dimensional system thermal–hydraulics codes will be used perhaps for a couple of decades from now. Computational Fluid Dynamics (CFD) tools are accepted at present to be a support of such analyses and they are used coupled to systems codes or as separate analysis tools for isolated components with boundary conditions obtained from systems codes. The restricted acceptance of “pure” CFD codes is due to many reasons but two of them are relevant, namely (a) the apparent lack of CFD grade experimental data and (b) the need for a complete verification and validation (V&V) and the uncertainty quantification for the codes currently available. There is plenty of experimental data related to integral test facilities (ITFs) that constitute macroscopic systems behavior information and a consolidated data base for such purposes. Despite of this, additional verification cases may be added to the above mentioned consolidated data. In the present paper, flow oscillations in parallel channel configurations with system codes are studied in diverse configurations. Different models, calculation options and, in particular, in-phase or out-of-phase oscillations were studied, both in heated and cooled parallel channels. The emphasis is on the effects of concentrated irreversible pressure losses coefficients at the inlet and at the outlet of the channels. In the case of cooled steam generator channels, the results of the Semiscale Integral Test Facility operating in natural circulation conditions are revisited. The results presented in this paper, show how a validation case lead to find a not still reported (in the Authors knowledge) verification case. The problem is related to twin-parallel-boiling and condensing, inverted U-tubes channels and connected through common plena. This is, of course, a problem that deserved many tens of papers in the last four decades. Flow splitting without reversal was computationally found and to explain this behavior a theoretical model limited in scope was developed that was a posteriori verified using a particular systems code (RELAP5) commonly applied to perform safety analyses of nuclear power plants. The rationale followed, the theoretical analysis performed and the confirmatory computational results found are summarized in this paper.

In a recently published (motivation) paper Lange et al., 2013 we analysed the BWR mode oscillation phenomenon from the bifurcation theory point of view. We demonstrated that some stability properties can be explained only in terms of nonlinear dynamics. In the present paper we shall continue the application of elements of the nonlinear stability analysis and analyse stability states which are characterized by the existence of simultaneous Hopf bifurcations (so-called Hopf–Hopf-bifurcations). From the bifurcation theory it is well-known that the solution manifold of the DEs describing the BWR dynamics can be surprisingly extensive in the vicinity of such bifurcation points. In the present paper we analyse the dynamics of a real BWR and search for a Hopf–Hopf point by appropriate parameter variation. The objective of this paper is to investigate and demonstrate the system dynamics which we have to expect if a Hopf–Hopf bifurcation exists. In a future paper we will analyse these different stability states reflecting the solution manifold near a Hopf–Hopf point concerning their technical relevance.

This paper deals with the theory of higher order modes in the Feynman Y function and the cross power spectral density (CPSD) in an accelerator-driven system (ADS) where pulsed spallation neutrons are injected at a constant frequency. A theoretical formula that considers the effect of higher order modes of the correlated and uncorrelated components in the Feynman Y function for an ADS is newly derived in this paper. Although the formula for the correlated and uncorrelated components of a CPSD was derived in a recent publication of the author of this paper, the verification has not been done yet. These formulae for the Feynman Y function and CPSD are applied to a subcritical multiplying system with a one-dimensional infinite slab geometry. The Feynman Y functions and CPSD calculated with the theoretical formulae are compared with the Monte Carlo simulations of these noise techniques. The theoretical formulae reproduce the Monte Carlo simulations very well, thereby substantiating the theoretical formulae derived in this paper. The correlated and uncorrelated components of the Feynman Y functions and CPSD are decomposed into the sum of the fundamental mode and other higher order modes. The higher order mode effect in the uncorrelated component is much more significant than in the correlated component.

In this paper, the application of goal based error measures for anisotropic adaptivity applied to shielding problems in which a detector is present is explored. Goal based adaptivity is important when the response of a detector is required to ensure that dose limits are adhered to. To achieve this, a dual (adjoint) problem is solved which solves the neutron transport equation in terms of the response variables, in this case the detector response. The methods presented can be applied to general finite element solvers, however, the derivation of the residuals are dependent on the underlying finite element scheme which is also discussed in this paper. Once error metrics for the forward and adjoint solutions have been formed they are combined using a novel approach. The two metrics are combined by forming the minimum ellipsoid that covers both the error metrics rather than taking the maximum ellipsoid that is contained within the metrics. Another novel approach used within this paper is the construction of the residual. The residual, used to form the goal based error metrics, is calculated from the subgrid scale correction which is inherent in the underlying spatial discretisation employed.

This paper describes an adjoint-based a posteriori error method for improving computational estimates of a functional. Examples of a functional occur in many areas of physics and engineering; they include lift and drag past an obstacle in aircraft and ship design and glacier movement in geology. In nuclear physics they include quantities such as power in a reactor fuel pin, the K eff eigenvalues in a criticality problem, or radiation input to a shield or the response of a detector. This paper shows how to use the adjoint system of equations to derive an error measure for improving these types of functional. The approach is to compute a reliable approximation to the error contained within the functional. A novel scheme for locally enriching the solution in order to obtain a higher order solution is used to define this approximation to the error, which is then subtracted from the functional to improve it. This goal-based scheme also has potential use in automating mesh adaption and is well aligned, therefore, with current work of other groups. The scheme developed in this paper offers an alternative to adapting the mesh in order to improve a functional.

Spectral elements methods provide very accurate solutions of elliptic problems. In this paper we apply the method to the A N (i.e. SP 2 N −1 ) approximation of neutron transport. Numerical results for classical benchmark cases highlight its performance in comparison with finite element computations, in terms of accuracy per degree of freedom and convergence rate. All calculations presented in this paper refer to two-dimensional problems. The method can easily be extended to three-dimensional cases. The results illustrate promising features of the method for more complex transport problems.

In this paper a development of the HEXNEM nodal expansion methods is presented, together with a program implementation and results from comparisons with benchmark problem solutions. The new development consists in the introduction of tangentially weighted exponential basis functions and continuity and boundary conditions for the tangential moments of the partial current in a way similar to that employed in some methods of the AFEN family. This is aimed at a better accounting for the flux variation along the radial node faces which for the large WWER-1000 assemblies can be a significant source of error. The test problem results presented in the paper indicate that an improved accuracy for WWER-1000 is achieved as compared with the HEXNEM2 method.

The paper introduces the application of Gaussian Processes (GPs) to determine the subcriticality level in accelerator-driven systems (ADSs) through the interpretation of pulsed experiment data. ADSs have peculiar kinetic properties due to their special core design. For this reason, classical – inversion techniques based on point kinetic (PK) generally fail to generate an accurate estimate of reactor subcriticality. Similarly to Artificial Neural Networks (ANNs), Gaussian Processes can be successfully trained to learn the underlying inverse neutron kinetic model and, as such, they are not limited to the model choice. Importantly, GPs are strongly rooted into the Bayes’ theorem which makes them a powerful tool for statistical inference. Here, GPs have been designed and trained on a set of kinetics models (e.g. point kinetics and multi-point kinetics) for homogeneous and heterogeneous settings. The results presented in the paper show that GPs are very efficient and accurate in predicting the reactivity for ADS-like systems. The variance computed via GPs may provide an indication on how to generate additional data as function of the desired accuracy.

This paper explores the application of the inner element subgrid scale method to the Boltzmann transport equation using quadratic basis functions. Previously, only linear basis functions for both the coarse scale and the fine scale were considered. This paper, therefore, analyses the advantages of using different coarse and subgrid basis functions for increasing the accuracy of the subgrid scale method. The transport of neutral particle radiation may be described by the Boltzmann transport equation (BTE) which, due to its 7 dimensional phase space, is computationally expensive to resolve. Multi-scale methods offer an approach to efficiently resolve the spatial dimensions of the BTE by separating the solution into its coarse and fine scales and formulating a solution whereby only the computationally efficient coarse scales need to be solved. In previous work an inner element subgrid scale method was developed that applied a linear continuous and discontinuous finite element method to represent the solution’s coarse and fine scale components. This approach was shown to generate efficient and stable solutions, and so this article continues its development by formulating higher order quadratic finite element expansions over the continuous and discontinuous scales. Here it is shown that a solution’s convergence can be improved significantly using higher order basis functions. Furthermore, by using linear finite elements to represent coarse scales in combination with quadratic fine scales, convergence can also be improved with only a modest increase in computational expense.

In this paper, a whole-core stochastic–deterministic hybrid coarse mesh transport method is extended to 2-D hexagonal geometry. This method may be used to calculate the eigenvalue and explicit pin fission density profile of hexagonal reactor cores. It models the exact detail within complex heterogeneous cores without homogenizing regions or materials, and neither block-level nor core-level asymmetry poses any limitations to the method. It solves eigenvalue problems by first splitting the core into a set of coarse meshes, and then using Monte Carlo methods to create a library of response expansion coefficients, found by expanding the angular current in phase-space distribution using a set of polynomials orthogonal on the angular half-space defined by mesh boundaries. The coarse meshes are coupled by the angular current at their interfaces. A deterministic sweeping procedure is then used to iteratively construct the solution. The method is evaluated using benchmark problems based on a gas-cooled, graphite-moderated high temperature reactor. The method quickly solves problems to any level of detail desired by the user. In this paper, it is used to explicitly calculate the fission density of individual fuel pins and determine the reactivity worth of individual control rods. In every case, results for the core multiplication factor and pin fission density distribution are found within several minutes. Results are highly accurate when compared to direct Monte Carlo reference solutions; errors in the eigenvalue calculations are on the order of 0.02%, and errors in the pin fission density average less than 0.1%.

Recently, a method was developed to generalize the multigroup theory to estimate the fine-group angular flux within a coarse-group transport calculation. In the development of that method, the angular dependence of the coarse-group total cross section was neglected. As is well known, this approximation introduces errors in the transport solution making its accuracy dependent on the number of coarse groups and the choice of group structure. This paper extends the generalized energy condensation theory to explicitly account for the angular dependence of the coarse-group total cross section. This is accomplished in a natural way by modifying the treatment of the total cross section to include orthogonal expansions in both energy and angle. As a result, the fine-group flux can be consistently reproduced during the coarse-group calculation. This method paves the way for recondensation of the cross sections on-the-fly thereby eliminating the errors introduced by using simplified problems (e.g. lattice-cells) to estimate the coarse-group cross sections. In this paper, the method is derived in general geometry and implemented and verified with several 1D reactor problems (both LWR and VHTR).

Higher modes of the neutron diffusion equation are required in some applications such as second order perturbation theory, and modal kinetics. In an earlier paper we had discussed a method for computing the α-modes of the diffusion equation. The discussion assumed that all neutrons are prompt. The present paper describes an extension of the method for finding the α-modes of diffusion equation with the inclusion of delayed neutrons. Such modes are particularly suitable for expanding the time dependent flux in a reactor for describing transients in a reactor. The method is illustrated by applying it to a three dimensional heavy water reactor model problem. The problem is solved in two and three neutron energy groups and with one and six delayed neutron groups. The results show that while the delayed α-modes are similar to λ-modes they are quite different from prompt modes. The difference gets progressively larger as we go to higher modes.

In this paper, the hierarchical domain decomposition boundary element method (HDD-BEM), which has been developed to solve the diffusion equation, is applied to the simplified P3 (SP3) equation. The HDD-BEM solution for the SP3 equation is provided in the present paper. A computer program, ABEMIE, based on the HDD-BEM is developed, and a two-dimensional one-group anisotropic-scattering benchmark problem is solved with it to verify the present HDD-BEM for the SP3 equation. Through numerical benchmarking, it is shown that the present method results in good agreement with the solution obtained using the existing SPN solver based on the finite element method for both eigenvalue and neutron flux distribution. This benchmark result suggests that the HDD-BEM is suitable for application to the SPN equation.

The fractional point-neutron kinetics model for the dynamic behavior in a nuclear reactor is derived and analyzed in this paper. The fractional model retains the main dynamic characteristics of the neutron motion in which the relaxation time associated with a rapid variation in the neutron flux contains a fractional order, acting as exponent of the relaxation time, to obtain the best representation of a nuclear reactor dynamics. The physical interpretation of the fractional order is related with non-Fickian effects from the neutron diffusion equation point of view. The numerical approximation to the solution of the fractional neutron point kinetics model, which can be represented as a multi-term high-order linear fractional differential equation, is calculated by reducing the problem to a system of ordinary and fractional differential equations. The numerical stability of the fractional scheme is investigated in this work. Results for neutron dynamic behavior for both positive and negative reactivity and for different values of fractional order are shown and compared with the classic neutron point kinetic equations. Additionally, a related review with the neutron point kinetics equations is presented, which encompasses papers written in English about this research topic (as well as some books and technical reports) published since 1940 up to 2010.

This paper discusses the use of the dimension-wise expansion model for cross-section parameterization.The components of the model were approximated with tensor products of orthogonal polynomials. As wedemonstrate, the model for a specific cross-section can be built in a systematic way directly from datawithout any a priori knowledge of its structure. The methodology is able to construct a finite basis oforthogonal polynomials that is required to approximate a cross-section with pre-specified accuracy.The methodology includes a global sensitivity analysis that indicates irrelevant state parameters whichcan be excluded from the model without compromising the accuracy of the approximation and withoutrepetition of the fitting process. To fit the dimension-wise expansion model, Randomised Quasi-Monte-Carlo Integration and Sparse Grid Integration methods were used. To test the parameterizationmethods with different integrations embedded we have used the OECD PBMR 400 MW benchmarkproblem. It has been shown in this paper that the Sparse Grid Integration achieves pre-specified accuracywith a significantly (up to 1–2 orders of magnitude) smaller number of samples compared to RandomisedQuasi-Monte-Carlo Integration. 2010 Elsevier Ltd. All rights reserved.

Coarse-mesh numerical methods are very efficient in the sense that they generate accurate results in short computational time, as the number of floating point operations generally decrease, as a result of the reduced number of mesh points. On the other hand, they generate numerical solutions that do not give detailed information on the problem solution profile, as the grid points can be located considerably away from each other. In this paper we describe two steps for the analytical reconstruction of the coarse-mesh solution generated by the spectral nodal method for neutral particle discrete ordinates ( S N ) transport model in slab geometry. The first step of the algorithm is based on the analytical reconstruction of the coarse-mesh solution within each discretization cell of the grid set up on the spatial domain. The second step is based on the angular reconstruction of the discrete ordinates solution between two contiguous ordinates of the angular quadrature set used in the S N model. Numerical results are given so we can illustrate the accuracy of the two reconstruction techniques, as described in this paper.

A novel technique for estimating system parameters based on consistent output estimate in wavelet domain is proposed as an alternative solution to the classical least squares minimization problem, in this paper. The algorithm is computationally efficient and exhibits excellent performance in cross validation. An iterative algorithm, alternately projecting the solution in time and wavelet domain for penalized minimization of local error in wavelet coefficients, is presented in the paper. As a case study, the paper addresses the problem of modeling a complex process in a large Pressurized Heavy Water Reactor (PHWR) based on the evolution of input and output. Xenon-induced spatial oscillations are kept from growing in large PHWRs, by means of the Liquid Zone Control System (LZCS). In this work, an identification scheme of a single input single output linear time invariant model of the LZCS is studied. Shift invariant orthogonal wavelets are used for consistent output estimate of the LZCS process. The technique is verified on the real experimental data obtained from the LZCS test set-up. It is observed that even a low order wavelet based model gives excellent approximation from fewer numbers of input–output data points.

This paper is concerned with the Indian design of a 220 MWe pressurized heavy water reactor (PHWR) having natural uranium (NU) fuel and heavy water as moderator and coolant. At the beginning of life, it is necessary to flatten the power by loading some depleted uranium (DU) bundles to achieve a nearly full power operation. The determination of best possible locations of DU bundles, which maximize fuel economy as well as safety, is a large-sized combinatorial optimization problem with constraints. In the past, 384 DU bundles have been loaded in locations determined by manual intuition in an Indian PHWR and maximum permissible power of 93% full power (FP) was obtained. In the present paper, a modern evolutionary algorithm called estimation of distribution algorithm (EDA) is used to improve upon this distribution. Optimum distributions of DU bundles which maximize K eff and give 100% FP without violating safety parameters such as maximum permissible bundle power, channel power, channel outlet temperature and permitted reactivity worths of shut-down systems are obtained. Another aspect studied in this paper is to find out how far one can increase the number of DU bundles loaded in the core. This will minimize the NU bundles requirement, extract more power from DU bundles and thus provide better fuel utilization. The idea is to conserve NU bundles. The optimum distribution of DU bundles has been obtained for the total number of DU bundles ranging from a few hundreds to a few thousands. It is found that, depending on various conditions, about 60–80% of the core can be loaded with DU bundles leading to a substantial saving in NU bundles. Some variation in the implementation of EDA to generate loading pattern of PHWR reactor core is also studied.

In this paper, analytical expressions for the Rossi- α and the Feynmann Y functions are deduced for the case of Poissonian and non-Poissonian neutron sources when the stochastic pulsing method is used. These analytical expressions are used to fit the experimental data and to obtain the prompt neutron time constant. Also we perform in this paper a comparison of the results obtained for the Rossi- α and Feynmann Y functions with Poissonian and non-Poissonian neutron sources, and we study how much change the shape of these functions when the fission probability decreases and the capture probability increases due to the depletion with time of the fuel, and the increase of the fission products. Some comparisons with experimental data and with the results of other authors have been performed. Another important question analyzed in this paper and that it is interesting from an academic point of view is that the average number of detected counts induced by one single neutron injected in the system at an arbitrary time t ′, should obey in point kinetics theory an adjoint equation in the time domain. Also the cross-factorial moment of the number of counts induced by one neutron in two counting intervals should obey also an adjoint equation in the time domain with a source term that depends on the first moments. These results are a consequence of more general results that have been obtained using stochastic transport theory for the one particle probability generating function or Kernel generating function.

In this paper a framework is derived based on first order perturbation theory to calculate the sensitivity of a transient in a nuclear reactor to delayed neutron parameters. The approach is based on the Adjoint Sensitivity Analysis Procedure as outlined in Cacuci [Cacuci, D., 2003. Sensitivity and Uncertainty Analysis. Theory, vol. I. CRC Press]. In this paper, the required adjoint system is defined and discussed, and an analytical solution is provided for a simplified case. Application of the ASAP requires that the time-dependent adjoint for neutrons and precursors is solved, and the implementation of such a solver is discussed. Proof-of-concept calculations have been performed for a 0D and a 3D case. It is found that sensitivities for the delayed neutron data are generally low. For increasing transients, the sensitivities increase with reactivity, with the highest sensitivities occuring for the delay groups with largest delayed neutron fraction β k , and for the largest ratio λ k / β k . For decreasing transients, the trends are less clearly defined, with some sensitivities increasing, and others decreasing, for increasingly negative reactivity. The proof-of-concept calculations have shown that the ASAP method yields good accuracy, but especially for 2D or 3D problems the method requires the use of advanced ordinary differential equation (ODE) solvers with built-in sensitivity capabilities, to keep the memory requirements and computational overhead in manageable bounds.

The zero-power reactor noise theory in a steady neutron multiplying medium was extended recently to a medium randomly varying in time to bridge the fields of the zero-power and the power reactor noise. For a time-varying medium in which the transition probability randomly fluctuates, only the use of the probability generating function technique based on the forward master equation approach is practical. However, with the forward master equation approach, the treatment of the joint moments of the neutron number and the medium state leads to a closure problem. Recently, the capability of the moment calculation technique for such cases was extended such that the closure problem could be solved exactly. The present paper describes and demonstrates this closure-free moment calculation technique in a time-varying binary multiplying medium, in which the medium state has two possible realizations. In addition to the first two moments of the neutron number N alone (irrespective of the medium state η), the joint moments of Nn and ηm, i.e., 〈Nnηm〉, were also obtained in a compact form for n = 1, 2 and arbitrary values of m, without a closure assumption. It was found that, for even m values, the asymptotic values of Nn and ηm are uncorrelated, whereas, for odd m values, they are negatively correlated, namely, their covariance is less than zero. The first two moments of the neutron number theoretically obtained were verified by the Monte Carlo technique. A perfect agreement was found between the Monte Carlo and the theoretical solutions. The closure-free moment calculation technique demonstrated in the present paper is expected to be applicable to various other problems related to the birth-and-death process with fluctuations of the transition probability, in which a closure problem occurs.

The third order neutron correlation technique is one of the subcriticality measurement techniques. This technique was originally proposed by Furuhashi and utilizes the second and third order neutron correlation factors Y(T) and X(T), which are evaluated from the variance and third order central moment of neutron counts detected during a counting gate width T, respectively. We can obtain the absolute value of subcriticality from the Y∞ and X∞, which are saturation values of Y(T) and X(T) when T goes to infinity. In the previous paper, we derived the generalized theoretical formulas of Y∞ and X∞ that took account of both the spatial and neutron energy effects. Its derivation was based on a heuristic method in which we consider the branching process of neutron family, and we utilized the α-eigenfunction expansion technique. Then, the previous formulas are expressed by multiple sums involving both the α-eigenfunctions and their adjoint functions. In this paper, we derive the new compact expressions of Y∞ and X∞ by using the importance functions related to the neutron detection process. Present derivation is based on the stochastic equation of neutron transport. New expressions have two advantages. We can clarify the physical meanings of Y∞ and X∞. And, without α-eigenfunction expansion, we can calculate Y∞ and X∞ directly by calculating the importance functions which satisfy the adjoint neutron transport equations. Moreover, we prove that new expressions of Y∞ and X∞ are identical with the previous ones, so that we can conclude that our previous derivation based on a heuristic method is correct.

This paper presents some new investigations on the use of Orthomin(1) algorithm for the solution of K -eigenvalue problem in Nuclear Reactor Physics. The algorithm was first used by Suetomi and Sekimoto [Conjugate gradient like methods and their application to eigenvalue problems for neutron diffusion equations, Annals of Nuclear Energy, 18(4) (1991) 205–227.] to find fundamental mode solution of K -eigenvalue problem based on multi-group neutron diffusion theory. It was shown to be an efficient alternative to the conventional methods based on power iterations. The use of this algorithm needs software development to evaluate product of coefficient matrices with vectors. Here, it is shown that the algorithm can also be applied to the K -eigenvalue equation cast in terms of fission matrix. This opens up the possibility of using conventional well-established diffusion codes to implement Orthomin(1) so that new software is not needed. Apart from this, a remarkable feature of Orthomin(1) brought out in this paper is its utility to find higher mode solutions of the K -eigenvalue problem. This procedure is radically different from the methods commonly used for finding higher K -modes. It may also be of interest for finding higher mode eigensolutions in other scientific fields.

In this paper, two new wavelet bases are developed for discretising the angular term of the first-order Boltzmann transport equation. The wavelets proposed are based on Sweldens second generation wavelets [Sweldens, W., 1993. The lifting scheme: a construction of second generation wavelets. SIAM J. Math. 1, 54], which are constructed through the lifting procedure [Sweldens, W., 1995. The lifting scheme: a new philosophy in biorthogonal wavelet construction. Wavelet Applications in Signal and Image Processing III]. In this paper, the wavelets are built on an octahedral domain, Fig. 2, and the angular flux approximation takes the form of finite element linear and quadratic representations. Full details of the meshing over the octahedron and derivation of the wavelet functions are given. The wavelets discussed are similar to the wavelets developed in Buchan [Buchan, A., 2003 c. Angular discretisation of the first order Boltzmann transport equation. Part 2: linear spherical wavelets. Technical Report, Imperial College, London, Dep. Earth Sci. Eng.] and [Buchan, A., 2003b. Angular discretisation of the first order Boltzmann transport equation. Part 3: quadratic spherical wavelets. Technical Report, Imperial College, London, Dep. Earth Sci. Eng.], in this paper the bases use a new fundamental amendment for mitigating the inaccuracies observed with the earlier bases. The performance of the new angular discretisation techniques are demonstrated using 2 one-dimensional and 4 two-dimensional test problems. These problems demonstrate the accuracy and susceptibility to ray effects of the proposed methods. Comparisons of all calculations are made with the conventional S N and P N approximations. Benchmark solutions are provided by the established code EVENT.

In complex and risky plants, such as the nuclear reactors, the analysis of the signals released by the many sensors which monitor the plant represents a difficult task due to the high-dimensionality of the data. This paper is the first of two in which we tackle the problem of the dimensionality reduction by the nonlinear principal components analysis as performed by an autoassociative neural network (AANN). This network filters the many input data and releases at the bottleneck output a relatively small number of signals which capture the significant properties of the original data, thus realizing the data reduction. In the present paper, we show that the network ability in correctly reproducing as output the given input after a passage through the bottleneck layer (which by definition should have fewer nodes than either input or output layers) could be conceived as a topological mapping between abstract spaces. Apart from the less critical choice of the number of nodes in the mapping and demapping layers, the topological mapping will be successful – and the AANN will be able to perform the required data reconstruction – provided that the number of nodes of the bottleneck layer is related to the dimensionality d of the abstract projection space. We show how to obtain a numerical estimate d* for the real dimension d. This numerical estimate will firmly base the choice of the number of nodes f of the bottleneck layer, thus avoiding the usual troubling trial-and-error procedure. The power of the proposed approach is demonstrated firstly on a few geometrical cases and then on the analysis of nuclear transients simulated by the classic Chernick’s model.

An effective method of calculating the pulsed Feynman-alpha formula for finite width pulses is introduced and applied in this paper. The method is suitable for calculating both the deterministic and the stochastic Feynman-alpha formulae, while also being capable of treating various pulse shapes through very similar steps and partly identical formulae. In the paper both the deterministic and the stochastic cases are treated for square and Gaussian pulses. The solutions show a very good agreement with the results of currently performed experiments by some of the authors at the Kyoto University Critical Assembly (KUCA). The formulae obtained are also used for a quantitative evaluation of the prompt neutron decay constant from a large number of experiments made at the KUCA for a wide range of parameters such as subcritical reactivity, pulse repetition frequency and pulse width. The suitability of the formulae to determine the prompt neutron decay constant a by curve fitting to the measured data was investigated. It was found that, despite the larger deviation from the traditional Feynman Y(T)-curves from the traditional ones with a constant source (i.e., larger ripples superimposed on a smooth curve), the stochastic pulsing method is superior to the deterministic one in that it yields the correct a value for all subcriticalities. The deterministic method also works fine for most cases, but its application is not so straightforward.

In this paper, performances of spatial homogenization methods for fuel or non-fuel cells are compared in slab geometry in order to facilitate pin-by-pin core calculations. Since the spatial homogenization methods were mainly developed for fuel assemblies, systematic study of their performance for the cell-level homogenization has not been carried out. Importance of cell-level homogenization is recently increasing since the pin-by-pin mesh core calculation in actual three-dimensional geometry, which is less approximate approach than current advanced nodal method, is getting feasible. Four homogenization methods were investigated in this paper; the flux-volume weighting, the generalized equivalence theory, the superhomogenization (SPH) method and the nonlinear iteration method. The last one, the nonlinear iteration method, was tested as the homogenization method for the first time. The calculations were carried out in simplified colorset assembly configurations of PWR, which are simulated by slab geometries, and homogenization performances were evaluated through comparison with the reference cell-heterogeneous calculations. The calculation results revealed that the generalized equivalence theory showed best performance. Though the nonlinear iteration method can significantly reduce homogenization error, its performance was not as good as that of the generalized equivalence theory. Through comparison of the results obtained by the generalized equivalence theory and the superhomogenization method, important byproduct was obtained; deficiency of the current superhomogenization method, which could be improved by incorporating the “cell-level discontinuity factor between assemblies”, was clarified.

This paper starts with some subjective reflections on the properties, interpretation and the use of the adjoint equation in neutron transport theory. This is used as a setting to put into perspective the seminal contribution of Hugo van Dam to the development of neutron noise diagnostics by introducing the use of the dynamic adjoint function. The significance of this development, although clear to most researchers in the field, has not been discussed formally and proper credit has not been given to its significance. It is the intention of this paper to make up for this fact. Some personal reminiscences on other areas of Hugo's scientific activity are also given at the end.