## Publications

89 results found

Ibekwe R, Cooling C, Trainer A,
et al., 2020, Modelling the short-term and long-term behaviour of the Oklo natural nuclear reactor phenomenon, *Progress in Nuclear Energy*, Vol: 118, ISSN: 0149-1970

This paper presents a computational investigation of the short-term and long-term behaviour of the Oklo natural nuclear reactors, instances in the distant past in which natural uranium deposits developed self-sustaining nuclear chain reactions. For the first time, processes occurring on timescales of seconds (such as changing temperature, moderator availability and power) are coupled in a single simulation with processes occurring over timescales of thousands of years (such as changing enrichment, reactor geometry and isotopic composition). This simulation reproduces key features of the Oklo reactors found in the literature (the cyclic boiling and flow of water in and out of the reactor; the characteristic three-hour cycle time; the total energy released by the reaction), gives greater insight into their development and evolution, and demonstrates a non-cyclic, non-boiling regime of behaviour in the later stages of reactor operation that has not previously been described.

Williams A, Williams M, Eaton M, 2019, Methods of studying the effect of rough surfaces on reactivity in two-dimensional reactor physics problems, *Annals of Nuclear Energy*, Vol: 130, Pages: 493-511, ISSN: 0306-4549

Rough and perturbed surfaces are examined in the field of reactor physics using homotopy, homogenisation and the Feinberg-Galanin method. The homotopy method allows a problem with a perturbed interface to be represented as an unperturbed problem with a modified boundary condition. Perturbation theory was applied to this method and the results were studied for the neutron transport equation, the neutron diffusion approximation and a hybrid ABH-diffusion approximation. The homogenisation of surface roughness in the fuelmoderator interface was also examined and compared to the homotopy approach. Finally, the effect of larger-scale geometric uncertainties was studied using the Feinberg-Galanin approach. The effectiveness of the methods is determined by examining the change in effective multiplication factor keff examined both directly and via the six-factor formula. Analytic solutions to approximations of perturbation theory differentials were capable ofaccurately predicting the change in keff when compared to numerical solutions. Examining a change in fuel pin radius of 20 µm, we find that for water moderated systems the overall change in eff k is close to 70 pcm and for graphite systems it is 18 pcm. This shows that water and graphite moderated cores are sensitive to small changes in geometry.

Perrier H, Denner F, Eaton MD,
et al., 2019, On the numerical modelling of Corium spreading using Volume-of-Fluid methods, *NUCLEAR ENGINEERING AND DESIGN*, Vol: 345, Pages: 216-232, ISSN: 0029-5493

Duan Y, Jackson C, Eaton M,
et al., 2019, An assessment of eddy viscosity models on predicting performance parameters of valves, *Nuclear Engineering and Design*, Vol: 342, Pages: 60-77, ISSN: 0029-5493

The major objective of the present study is to evaluate the performance of a range of turbulent eddy viscosity models in the prediction of macro-parameters (flow coefficient (CQ) and force coefficient (CF)), for certain types of valve, including the conic valve, the disk valves, and the compensated valve. This has been achieved by comparison of numerical predictions with experimental measurements available in the literature. The examined turbulence models include most of the available turbulent eddy viscosity models in STAR-CCM+ 12.04. They are the standard k-ε model, realizable k-ε model, k-ω-sst model, V2F model, EB k-ε model and the Lag EB k-ε models.The low-Re turbulence models (k-ω-sst, V2F, EB k-ε and Lag EB k-ε) perform worse than the high-Re models (standard k-ε and realizable k-ε). For the conic valve, the performance of different turbulent models varies little; the standard k-ε model shows a marginal advantage over the others. The performance of the turbulence models changed greatly, however, for prediction of CQ and CF of the disk and compensated valves. In general, the realizable k-ε model is demonstrated to be a robust choice for both valve types. Although the EB k-ε may marginally outperform it in the prediction of CF at large disk valve opening.The effects of the unknown entry flow and initialization conditions are also studied. The predictions are more sensitive to the entry flow condition when the valve opening is large. Additionally, the uncertainties caused by unknown entry conditions are comparable to overall modelling errors in some cases. For flow systems with multiple stable flow-states coexisting in the flow domain, the output of the numerical models can also be affected by the initialization conditions.When the streamline curvature and secondary flow is modest like conical valve flow, the nonlinear modification of the standard k-ε mode

Winter G, Cooling C, Williams M,
et al., 2018, Importance of parametric uncertainty in predicting probability distributions for burst wait-times in fissile systems, *Annals of Nuclear Energy*, Vol: 119, Pages: 117-128, ISSN: 0306-4549

A method of uncertainty quantification in the calculation of wait-time probability distributions in delayed supercritical systems is presented. The method is based on Monte Carlo uncertainty quantification and makes use of the computationally efficient gamma distribution method for prediction of the wait-time probability distribution. The range of accuracy of the gamma distribution method is examined and parameterised based on the rate and magnitude of the reactivity insertion, the strength of the intrinsic neutron source and the prompt neutron lifetime. The saddlepoint method for inverting the generating function and a Monte Carlo simulation are used as benchmarks against which the accuracy of the gamma distribution method is determined. Finally, uncertainty quantification is applied to models of the Y-12 accident and experiments of Authier et al. (2014) on the Caliban reactor.

O'Malley B, Kophazi J, Eaton MD,
et al., 2018, Pyramid finite elements for discontinuous and continuous discretizations of the neutron diffusion equation with applications to reactor physics, *Progress in Nuclear Energy*, Vol: 105, Pages: 175-184, ISSN: 0149-1970

When using unstructured mesh finite element methods for neutron diffusion problems, hexahedral elements are in most cases the most computationally efficient and accurate for a prescribed number of degrees of freedom. However, it is not always practical to create a finite element mesh consisting entirely of hexahedral elements, particularly when modelling complex geometries, making it necessary to use tetrahedral elements to mesh more geometrically complex regions. In order to avoid hanging nodes, wedge or pyramid elements can be used in order to connect hexahedral and tetrahedral elements, but it was not until 2010 that a study by Bergot established a method of developing correct higher order basis functions for pyramid elements. This paper analyses the performance of first and second-order pyramid elements created using the Bergot method within continuous and discontinuous finite element discretisations of the neutron diffusion equation. These elements are then analysed for their performance using a number of reactor physics benchmarks. The accuracy of solutions using pyramid elements both alone and in a mixed element mesh is shown to be similar to that of meshes using the more standard element types. In addition, convergence rate analysis shows that, while problems discretized with pyramids do not converge as well as those with hexahedra, the pyramids display better convergence properties than tetrahedra.

Jeffers RS, Kophazi J, Eaton MD,
et al., 2018, Goal-Based Error Estimation, Functional Correction, h, p and hp Adaptivity of the 1-D Diamond Difference Discrete Ordinate Method, *Journal of Computational and Theoretical Transport*, Vol: 46, Pages: 427-458, ISSN: 2332-4309

This paper uses local dual weighted residual (DWR) error indicators to flag cells for goal-based refinement in a 1-D diamond difference (DD) discretisation of the discrete ordinate (SN) neutron transport equations. Goal-orientated adaptive mesh refinement (GO-AMR) aims to produce a mesh that is optimal for a given goal or QoI (Quantity of Interest). h, p and hp refinement is implemented and applied to various test cases. A merit function is derived for the combined hp algorithm and is calculated for each refinement option within each flagged cell. The refinement option with the highest merit function is chosen for a given flagged cell. This paper also investigates the use of the DWR error estimation as a correction term for the originally calculated QoI. If error correction and GO-AMR are combined the DWR error indicators do not always give an optimal mesh for the corrected value of the QoI. Therefore, refinement indicators based on the error in the error correction term are used and tested in this work.

Duan Y, Eaton MD, Bluck MJ, et al., 2018, A VALIDATION OF CFD METHODS ON PREDICTING VALVE PERFORMANCE PARAMETERS, ASME Power Conference 2018 (POWER2018), Publisher: AMER SOC MECHANICAL ENGINEERS

Ahn JS, Bluck M, Eaton M, et al., 2018, A VALIDATION OF RELAP ON PREDICTING NUCLEAR POWER PLANT PHENOMENA, 26th International Conference on Nuclear Engineering (ICONE-26), Publisher: AMER SOC MECHANICAL ENGINEERS

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Duan Y, Eaton MD, Bluck MJ, et al., 2018, ASSESSMENTS OF DIFFERENT TURBULENCE MODELS IN PREDICTING THE PERFORMANCE OF A BUTTERFLY VALVE, 26th International Conference on Nuclear Engineering (ICONE-26), Publisher: AMER SOC MECHANICAL ENGINEERS

Lindley B, Allen D, Lillington J, et al., 2018, MODELLING AND SIMULATION ACTIVITIES IN SUPPORT OF THE UK NUCLEAR R&D PROGRAMME ON DIGITAL REACTOR DESIGN, 26th International Conference on Nuclear Engineering (ICONE-26), Publisher: AMER SOC MECHANICAL ENGINEERS

Ferguson J, Kophazi J, Eaton MD, 2018, POLYGONAL VIRTUAL ELEMENT SPATIAL DISCRETISATION METHODS FOR THE NEUTRON DIFFUSION EQUATION WITH APPLICATIONS IN NUCLEAR REACTOR PHYSICS, 26th International Conference on Nuclear Engineering (ICONE-26), Publisher: AMER SOC MECHANICAL ENGINEERS

Wilson SG, Kophazi J, Owens AR, et al., 2018, INTERIOR PENALTY SCHEMES FOR DISCONTINUOUS ISOGEOMETRIC METHODS WITH AN APPLICATION TO NUCLEAR REACTOR PHYSICS, 26th International Conference on Nuclear Engineering (ICONE-26), Publisher: AMER SOC MECHANICAL ENGINEERS

Latimer C, Kophazi J, Eaton MD, 2018, ISOGEOMETRIC MULTI-LEVEL ITERATIVE SOLUTION ALGORITHMS WITH APPLICATIONS IN NUCLEAR REACTOR PHYSICS, 26th International Conference on Nuclear Engineering (ICONE-26), Publisher: AMER SOC MECHANICAL ENGINEERS

Cooling CM, Williams MMR, Eaton MD, 2017, CALLISTO-SPK: A Stochastic Point Kinetics Code for Performing Low Source Nuclear Power Plant Start-up and Power Ascension Calculations, *Annals of Nuclear Energy*, Vol: 113, Pages: 319-331, ISSN: 0306-4549

This paper presents the theory and application of a code called CALLISTO which is used for performing NPP start-up and power ascension calculations. The CALLISTO code is designed to calculate various values relating to the neutron population of a nuclear system which contains a low number of neutrons. These variables include the moments of the PDF of the neutron population, the maturity time and the source multiplier. The code itself is based upon the mathematics presented in another paper and utilises representations of the neutron population which are independent of both space and angle but allows for the specification of an arbitrary number of energy groups.Five examples of the use of the code are presented. Comparison is performed against results found in the literature and the degree of agreement is discussed. In general the agreement is found to be good and, where it is not, plausible explanations for discrepancies are presented. The final two cases presented examine the effect of the number of neutron groups included and finds that, for the systems simulated, there is no significant difference in the key results of the code.

O'Malley B, Kophazi J, Eaton MD,
et al., 2017, Discontinuous Galerkin spatial discretisation of the neutron transport equation with pyramid finite elements and a discrete ordinate (SN) angular approximation, *Annals of Nuclear Energy*, Vol: 113, Pages: 526-535, ISSN: 0306-4549

In finite element analysis it is well known that hexahedral elements are the preferred type of three dimensional element because of their accuracy and convergence properties. However, in general it is not possible to mesh complex geometry problems using purely hexahedral meshes. Indeed for highly complex geometries a mixture of hexahedra and tetrahedra is often required. However, in order to geometrically link hexahedra and tetrahedra, in a conforming finite element mesh, pyramid elements will be required. Until recently the basis functions of pyramid elements were not fully understood from a mathematical or computational perspective. Indeed only first-order pyramid basis functions were rigorously derived and used within the field of finite elements. This paper makes use of a method developed by Bergot that enables the generation of second and higher-order basis functions, applying them to finite element discretisations of the neutron transport equation in order to solve nuclear reactor physics, radiation shielding and nuclear criticality problems. The results demonstrate that the pyramid elements perform well in almost all cases in terms of both solution accuracy and convergence properties.

Williams MMR, Eaton MD, 2017, Spatial Effects in Low Neutron Source Start-up and Associated Stochastic Phenomena, *Annals of Nuclear Energy*, Vol: 111, Pages: 616-634, ISSN: 0306-4549

This work concerns the calculation of the neutron source strength necessary to start up a nuclear reactor such that the likelihood of an undesirable stochastic transient is reduced to a specified value (e.g. 10−8). We extend our earlier point model work on low source calculations to include the spatial variation of the neutron source. Results for the source multiplier for a given safety factor are obtained for slab, cylindrical and spherical systems. The spatial term in the Pál-Bell equation is dealt with by Chebyshev-Gauss-Lobatto collocation methods and this enables an extrapolation distance to be included, thereby simulating a reflector. Results are given for a range of system sizes, and corresponding source multipliers for safe source determination are obtained. The saddlepoint method is used to invert the generating function. In addition to the low source calculations, we have also tested the collocation method on the survival probability in a sphere which demonstrates excellent convergence. We also comment on the usefulness of the Gamma pdf for spatially dependent problems. For clarity of presentation, some of the detailed mathematical work is relegated to Appendices.

Saxby JEM, Prinja A, Eaton M, 2017, Energy Dependent Transport Model of the Neutron Number Probability Distribution in a Subcritical Multiplying Assembly, *Nuclear Science and Engineering*, Vol: 189, Pages: 1-25, ISSN: 0029-5639

The time and phase-space dependent backward master equation is used to develop and numerically solve a coupled system of transport equations for the probability distribution of the neutron number in subregions of a spherically symmetric, reflected, subcritical plutonium sphere. The number distributions are computed for a single initial neutron injected into the assembly and localized in phase space as well as in the presence of a uniformly distributed spontaneous fission source in the fissile region. A standard multigroup, discrete ordinates scheme with second-order spatial and fully implicit time discretization proved sufficiently accurate for this application. The results presented show complex behaviors arising from the material interface and spectral effects due to neutron slowing down that cannot be encapsulated in a lumped model. Additionally, low-order spatial moments were computed both by averaging the number distributions of finite order and directly solving the transport equations for the moments using the same numerical scheme. While generally excellent agreement is observed between the two approaches, the truncation order has a noticeable effect on the accuracy of the higher moments that are computed using the number distributions.

Owens A, Kophazi J, Eaton M, 2017, Optimal trace inequality constants for interior penalty discontinuous Galerkin discretisations of elliptic operators using arbitrary elements with non-constant Jacobians, *Journal of Computational Physics*, Vol: 350, Pages: 847-870, ISSN: 0021-9991

In this paper, a new method to numerically calculate the trace inequality constants, which arise in the calculation of penalty parameters for interior penalty discretisations of elliptic operators, is presented. These constants are provably optimal for the inequality of interest. As their calculation is based on the solution of a generalised eigenvalue problem involving the volumetric and face stiffness matrices, the method is applicable to any element type for which these matrices can be calculated, including standard finite elements and the non-uniform rational B-splines of isogeometric analysis. In particular, the presented method does not require the Jacobian of the element to be constant, and so can be applied to a much wider variety of element shapes than are currently available in the literature. Numerical results are presented for a variety of finite element and isogeometric cases. When the Jacobian is constant, it is demonstrated that the new method produces lower penalty parameters than existing methods in the literature in all cases, which translates directly into savings in the solution time of the resulting linear system. When the Jacobian is not constant, it is shown that the naive application of existing approaches can result in penalty parameters that do not guarantee coercivity of the bilinear form, and by extension, the stability of the solution. The method of manufactured solutions is applied to a model reaction-diffusion equation with a range of parameters, and it is found that using penalty parameters based on the new trace inequality constants result in better conditioned linear systems, which can be solved approximately 11% faster than those produced by the methods from the literature.

Perrier H, Eaton M, Wachem BV, 2017, Volume of fluid simulation of isothermal viscous spreading flows, 17th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-17)

© 2016 Association for Computing Machinery Inc. All Rights Reserved. The spreading of an isothermal dense viscous liquid into a gas environment was investigated using Computational Fluid Dynamics (CFD) simulations. A Volume of Fluid (VOF) method was used to model the two-phase liquid/gas system. It was noticed that due to the use of VOF methods, a layer of gas could be trapped below the liquid front forming a “gas cushion”. This effect was described and a solution to prevent it using a Navier-Slip boundary condition at the wall was proposed. The influence of the Navier-Slip condition, numerical parameters, physical models and fluid properties on the spreading behavior was studied and simulation guidelines were proposed.

Welch J, Kophazi J, Owens A,
et al., 2017, A geometry preserving, conservative, mesh-to-mesh isogeometric interpolation algorithm for spatial adaptivity of the multigroup, second-order even-parity form of the neutron transport equation, *Journal of Computational Physics*, Vol: 347, Pages: 129-146, ISSN: 0021-9991

In this paper a method is presented for the application of energy-dependent spatial meshes applied to the multigroup, second-order, even-parity form of the neutron transport equation using Isogeometric Analysis (IGA). The computation of the inter-group regenerative source terms is based on conservative interpolation by Galerkin projection. The use of Non-Uniform Rational B-splines (NURBS) from the original computer-aided design (CAD) model allows for efficient implementation and calculation of the spatial projection operations while avoiding the complications of matching different geometric approximations faced by traditional finite element methods (FEM). The rate-of-convergence was verified using the method of manufactured solutions (MMS) and found to preserve the theoretical rates when interpolating between spatial meshes of different refinements. The scheme’s numerical efficiency was then studied using a series of two-energy group pincell test cases where a significant saving in the number of degrees-of-freedom can be found if the energy group with a complex variation in the solution is refined more than an energy group with a simpler solution function. Finally, the method was applied to a heterogeneous, seven-group reactor pincell where the spatial meshes for each energy group were adaptively selected for refinement. It was observed that by refining selected energy groups a reduction in the total number of degrees-of-freedom for the same total L2 error can be obtained.

Saxby JEM, prinja AK, Eaton MD, 2017, Diffusion theory model of the neutron number probability distribution in a subcritical multiplying assembly, *Annals of Nuclear Energy*, Vol: 109, Pages: 507-528, ISSN: 1873-2100

The probability distribution of neutron numbers in a symmetric subcritical reflected fissile sphere is numerically obtained using a one-speed diffusion approximation to the underlying backward Master equation. Employing an accurate space–time discretisation scheme, the coupled but closed system of equations for the number probabilities is sequentially solved as a function of position and time of an injected neutron. This solution is then used to construct the corresponding distributions for a random intrinsic source of arbitrary multiplicity. Numerical results clearly demonstrate the importance of including spatial dependence in the neutron number probability distributions, which show complex spatial behaviours that cannot be encapsulated in a point model system. This is especially evident in the case of the multi-region model where the presence of a reflector is seen to alter the approach to steady state of the number probabilities, while the material interface has a significant effect on the magnitude of the probabilities, both locally and globally.

O'Malley B, Kophazi J, Smedley-Stevenson RP,
et al., 2017, P-Multigrid expansion of hybrid multilevel solvers for discontinuous Galerkin finite element discrete ordinate (DG-FEM-SN) diffusion synthetic acceleration (DSA) of radiation transport algorithms, *Progress in Nuclear Energy*, Vol: 98, Pages: 177-186, ISSN: 0149-1970

Effective preconditioning of neutron diffusion problems is necessary for the development of efficient DSA schemes for neutron transport problems. This paper uses P-multigrid techniques to expand two preconditioners designed to solve the MIP diffusion neutron diffusion equation with a discontinuous Galerkin (DG-FEM) framework using first-order elements. These preconditioners are based on projecting the first-order DG-FEM formulation to either a linear continuous or a constant discontinuous FEM system. The P-multigrid expansion allows the preconditioners to be applied to problems discretised with second and higher-order elements. The preconditioning algorithms are defined in the form of both a V-cycle and W-cycle and applied to solve challenging neutron diffusion problems. In addition a hybrid preconditioner using P-multigrid and AMG without a constant or continuous coarsening is used. Their performance is measured against a computationally efficient standard algebraic multigrid preconditioner. The results obtained demonstrate that all preconditioners studied in this paper provide good convergence with the continuous method generally being the most computationally efficient. In terms of memory requirements the preconditioners studied significantly outperform the AMG.

Owens A, Welch J, Kophazi J,
et al., 2017, An adaptive, hanging-node, discontinuous isogeometric analysis method for the first-order form of the neutron transport equation with discrete ordinate (SN) angular discretisation, *Computer Methods in Applied Mechanics and Engineering*, Vol: 318, Pages: 215-241, ISSN: 0045-7825

In this paper a discontinuous, hanging-node, isogeometric analysis (IGA) method is developed and applied to the first-order form of the neutron transport equation with a discrete ordinate (SN) angular discretisation in two-dimensional space. The complexities involved in upwinding across curved element boundaries that contain hanging-nodes have been addressed to ensure that the scheme remains conservative. A robust algorithm for cycle-breaking has also been introduced in order to develop a unique sweep ordering of the elements for each discrete ordinates direction. The convergence rate of the scheme has been verified using the method of manufactured solutions (MMS) with a smooth solution. Heuristic error indicators have been used to drive an adaptive mesh refinement (AMR) algorithm to take advantage of the hanging-node discretisation. The effectiveness of this method is demonstrated for three test cases. The first is a homogeneous square in a vacuum with varying mean free path and a prescribed extraneous unit source. The second test case is a radiation shielding problem and the third is a 3×3 “supercell” featuring a burnable absorber. In the final test case, comparisons are made to the discontinuous Galerkin finite element method (DGFEM) using both straight-sided and curved quadratic finite elements.

Jeffers RS, Kópházi J, Eaton MD,
et al., 2017, Goal-based h-adaptivity of the 1-D diamond difference discrete ordinate method., *Journal of Computational Physics*, Vol: 335, Pages: 179-200, ISSN: 1090-2716

The quantity of interest (QoI) associated with a solution of a partial differential equation (PDE) is not, in general, the solution itself, but a functional of the solution. Dual weighted residual (DWR) error estimators are one way of providing an estimate of the error in the QoI resulting from the discretisation of the PDE.This paper aims to provide an estimate of the error in the QoI due to the spatial discretisation, where the discretisation scheme being used is the diamond difference (DD) method in space and discrete ordinate (SNSN) method in angle. The QoI are reaction rates in detectors and the value of the eigenvalue (Keff)(Keff) for 1-D fixed source and eigenvalue (KeffKeff criticality) neutron transport problems respectively. Local values of the DWR over individual cells are used as error indicators for goal-based mesh refinement, which aims to give an optimal mesh for a given QoI.

Owens A, Kophazi J, Welch J,
et al., 2017, Energy dependent mesh adaptivity of discontinuous isogeometric discrete ordinate methods with dual weighted residual error estimators, *Journal of Computational Physics*, Vol: 335, Pages: 352-386, ISSN: 1090-2716

Inthispaperahanging-node,discontinuousGalerkin,isogeometricdiscretisationofthemultigroup,discreteordinates (SN) equations is presented in which each energy group has its own mesh. The equations are discretised using NonUniform Rational B-Splines (NURBS), which allows the coarsest mesh to exactly represent the geometry for a wide rangeofengineeringproblemsofinterest; thiswouldnotbethecaseusingstraight-sidedﬁniteelements. Information istransferredbetweenmeshesviatheconstructionofasupermesh. Thisisanon-trivialtaskfortwoarbitrarymeshes, but is signiﬁcantly simpliﬁed here by deriving every mesh from a common coarsest initial mesh. In order to take full advantage of this ﬂexible discretisation, goal-based error estimators are derived for the multigroup, discrete ordinates equations with both ﬁxed (extraneous) and ﬁssion sources, and these estimators are used to drive an adaptive mesh reﬁnement (AMR) procedure. The method is applied to a variety of test cases for both ﬁxed and ﬁssion source problems. The error estimators are found to be extremely accurate for linear NURBS discretisations, with degraded performance for quadratic discretisations owing to a reduction in relative accuracy of the “exact” adjoint solution required to calculate the estimators. Nevertheless, the method seems to produce optimal meshes in the AMR process for both linear and quadratic discretisations, and is ≈ ×100 more accurate than uniform reﬁnement for the same amount of computational eﬀort for a 67 group deep penetration shielding problem.

Williams MMR, Eaton MD, 2017, A theory of low source start-up based on the Pál-Bell equations, *Annals of Nuclear Energy*, Vol: 102, Pages: 317-348, ISSN: 1873-2100

The safe start-up of a nuclear reactor depends upon the presence of a steady neutron source in the core. This source, however, does not always have to be physically inserted into the reactor because there exist in the core natural neutron sources from spontaneous fission, cosmic rays, photo neutrons, fission products, etc. Nevertheless, so that the source magnitude is well defined, it is generally thought judicious to have a specially constructed source of the (α,n) type present. From an operational point of view, it is vital to assess the strength of thenatural sources to see if they will be sufficient in magnitude to ensure safe stochastic startup without the addition of an extraneous source. The most important case for source evaluation is that of a reactor starting up with fresh, unirradiated fuel because then the natural background sources will be at a minimum. It is the purpose of this paper to examine the criteria necessary to ensure that the source strength is high enough to reduce the probability of any undesirable stochastic transient occurring to a specified value, e.g. 10e-8 ; it may also be considered as an update of the classic work of Hurwitz and co-workers (1963). To carry out the calculations, we use the Pál-Bell backward formalism (Pázsit and Pál, 2008) and apply it to the point model in order to make comparisons with the earlier work of Hurwitz. We also extend the study to include space and energy dependence which are found to have a not insignificant influence on the results. The usefulness of the Gamma distributionis explored and its accuracy assessed. Tables and figures are given to illustrate the conclusions.

O'Malley B, Kophazi J, Smedley-Stevenson RP,
et al., 2016, Hybrid multi-level solvers for discontinuous Galerkin finite element discrete ordinate (DG-FEM-SN) diffusion synthetic acceleration (DSA) of radiation transport algorithms, *Annals of Nuclear Energy*, Vol: 102, Pages: 134-147, ISSN: 1873-2100

his paper examines two established preconditioners which were developedto accelerate the solution of discontinuous Galerkin nite element method (DG-FEM) discretisations of the elliptic neutron di usion equation. They are eachpresented here as a potential way to accelerate the solution of the Modi ed In-terior Penalty (MIP) form of the discontinuous di usion equation, for use as adi usion synthetic acceleration (DSA) of DG-FEM discretisations of the neutrontransport equation. The preconditioners are both two-level schemes, di eringin the low-level space utilised. Once projected to the low-level space a selectionof algebraic multigrid (AMG) preconditioners are utilised to obtain a furthercorrection step, these are therefore \hybrid" preconditioners. The rst precon-ditioning scheme utilises a continuous piece-wise linear nite element method(FEM) space, while the second uses a discontinuous piece-wise constant space.Both projections are used alongside an element-wise block Jacobi smoother inorder to create a symmetric preconditioning scheme which may be used along-side a conjugate gradient algorithm. An eigenvalue analysis reveals that bothshould aid convergence but the piece-wise constant based method struggles withsome of the smoother error modes. Both are applied to a range of problems in-cluding some which are strongly heterogeneous. In terms of conjugate gradient(CG) iterations needed to reach convergence and computational time required,both methods perform well. However, the piece-wise linear continuous schemeappears to be the more e ective of the two. An analysis of computer memoryusage found that that the discontinuous piece-wise constant method had thelowest memory requirements.

Welch J, Kophazi J, Owens A,
et al., 2016, Isogeometric analysis for the multigroup neutron diffusion equation with applications in reactor physics, *Annals of Nuclear Energy*, Vol: 101, Pages: 465-480, ISSN: 1873-2100

Isogeometric Analysis (IGA) has been applied to heterogeneous reactor physics problems using the multigroup neutron dif-fusion equation. IGA uses a computer-aided design (CAD) description of the geometry commonly built from Non-UniformRational B-Splines (NURBS), which can exactly represent complicated curved shapes such as circles and cylinders, commonfeatures in reactor design. This work has focused on comparing IGA to nite element analysis (FEA) for heterogeneousreactor physics problems, including the OECD/NEA C5G7 LWR benchmark. The exact geometry and increased basisfunction continuity contribute to the accuracy of IGA and an improvement over comparable FEA calculations has beenobserved.

Cooling CM, Williams MMR, Eaton MD, 2016, Coupled probabilistic and point kinetics modelling of fast pulses in nuclear systems, *Annals of Nuclear Energy*, Vol: 94, Pages: 655-671, ISSN: 1873-2100

This paper describes a probabilistic method of modelling point nuclear systemswith low numbers of neutrons including the effects of delayed neutron precursors andits coupling with standard point kinetics equations. This coupling allows the simulationof the non-deterministic progression of a system transitioning from subcritical tosupercritical and the resulting power peak. Through analysis of large numbers of realisationsvarious statistical parameters of such transients can be obtained. The methodof simulation presented here successfully replicates the survival and extinction probabilitiespredicted by the Backwards Master Equation and experimental and analyticresults from the literature regarding the Godiva reactor and extends the examinationof that reactor. In particular the effect of delayed neutrons on the simulated responseof Godiva is highlighted. With its implementation in a parallel computer code, themodel is able to simulate at a reasonable speed a range of systems where low neutronpopulations are important.

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