## Publications

98 results found

Winter GE, Cooling CM, Eaton MD, 2020, Linear energy transfer of fission fragments of 235U and nucleation of gas bubbles in aqueous solutions of uranyl nitrate, *Annals of Nuclear Energy*, Vol: 142, Pages: 1-19, ISSN: 0306-4549

Fission fragments emitted in a fissile solution create tiny gas bubbles, the size of which is determined by the linear energy transfer (LET) of the particles. The LET of fission fragments of 235U in aqueous solutions of uranyl nitrate has been determined, and using methods adapted from the literature, the size of gas bubbles generated along the tracks of these particles has been estimated, revealing important variations with respect to particle LET and solution properties. Empirical correlations are presented for the maximum radius of radiolytic gas bubbles in unsaturated solutions of uranyl nitrate as a function of solution temperature and concentration. These can be used to predict the critical concentration of dissolved hydrogen necessary for the appearance of gas voids during nuclear criticality transients. The findings are intended for use in a future model of nuclear criticality transients in aqueous fissile solutions for the purposes of nuclear criticality safety assessment.

Williams MMR, Eaton MD, 2020, A theory of low source startup based on the Pal-Bell equations (vol 102, pg 317, 2017), *ANNALS OF NUCLEAR ENERGY*, Vol: 140, ISSN: 0306-4549

Williams MMR, Eaton MD, 2020, Spatial effects in low neutron source startup and associated stochastic phenomena (vol 111, pg 616, 2018), *ANNALS OF NUCLEAR ENERGY*, Vol: 140, ISSN: 0306-4549

Latimer C, Kópházi J, Eaton MD,
et al., 2020, A geometry conforming, isogeometric, weighted least squares (WLS) method for the neutron transport equation with discrete ordinate (SN) angular discretisation, *Progress in Nuclear Energy*, Vol: 121, Pages: 1-15, ISSN: 0149-1970

This paper presents the application of isogeometric analysis (IGA) to the spatial discretisation of the multi-group, source iteration compatible, weighted least squares (WLS) form of the neutron transport equation with a discrete ordinate (S) angular discretisation. The WLS equation is an elliptic, second-order form of the neutron transport equation that can be applied to neutron transport problems on computational domains where there are void regions present. However, the WLS equation only maintains conservation of neutrons in void regions in the fine mesh limit. The IGA spatial discretisation is based up non-uniform rational B-splines (NURBS) basis functions for both the test and trial functions. In addition a methodology for selecting the magnitude of the weighting function for void and near-void problems is presented. This methodology is based upon solving the first-order neutron transport equation over a coarse spatial mesh. The results of several nuclear reactor physics verification benchmark test cases are analysed. The results from these verification benchmarks demonstrate two key aspects. The first is that the magnitude of the error in the solution due to approximation of the geometry is greater than or equal to the magnitude of the error in the solution due to lack of conservation of neutrons. The second is the effect of the weighting factor on the solution which is investigated for a boiling water reactor (BWR) lattice that contains a burnable poison pincell. It is demonstrated that the smaller the area this weighting factor is active over the closer the WLS solution is to that produced by solving the self adjoint angular flux (SAAF) equation. Finally, the methodology for determining the magnitude of the weighting factor is shown to produce a suitable weighting factor for nuclear reactor physics problems containing void regions. The more refined the coarse solution of the first-order transport equation, the more suitable the weighting factor.

Kophazi J, Eaton M, McClarren R,
et al., 2020, A geometry conforming isogeometric method for the self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (SN) angular discretisation, *Annals of Nuclear Energy*, Vol: 136, Pages: 1-16, ISSN: 0306-4549

This paper presents the application of isogeometric analysis (IGA) to the spatial discretisationof the multi-group, self-adjoint angular flux (SAAF) form of the neutron transport equation witha discrete ordinate (SN) angular discretisation. The IGA spatial discretisation is based uponnon-uniform rational B-spline (NURBS) basis functions for both the test and trial functions. Inaddition a source iteration compatible maximum principle is used to derive the IGA spatiallydiscretised SAAF equation. It is demonstrated that this maximum principle is mathematicallyequivalent to the weak form of the SAAF equation. The rate of convergence of the IGA spatial discretisation of the SAAF equation is analysed using a method of manufactured solutions(MMS) verification test case. The results of several nuclear reactor physics verification benchmark test cases are analysed. This analysis demonstrates that for higher-order basis functions,and for the same number of degrees of freedom, the FE based spatial discretisation methods arenumerically less accurate than IGA methods. The difference in numerical accuracy between theIGA and FE methods is shown to be because of the higher-order continuity of NURBS basisfunctions within a NURBS patch as well as the preservation of both the volume and surfacearea throughout the solution domain within the IGA spatial discretisation. Finally, the numericalresults of applying the IGA SAAF method to the OECD/NEA, seven-group, two-dimensionalC5G7 quarter core nuclear reactor physics verification benchmark test case are presented. Theresults, from this verification benchmark test case, are shown to be in good agreement with solutions of the first-order form as well as the second-order even-parity form of the neutron transportequation for the same order of discrete ordinate (SN) angular approximation.

Jeffers RS, Kópházi J, Eaton MD,
et al., 2020, Goal-based error estimation for the multi-dimensional diamond difference and box discrete ordinate (SN) methods, *Journal of Computational and Theoretical Transport*, Vol: 49, Pages: 51-87, ISSN: 2332-4309

Goal-based error estimation due to spatial discretization and adaptive mesh refinement (AMR) has previously been investigated for the one dimensional, diamond difference, discrete ordinate (1-D DD-SN) method for discretizing the Neutron Transport Equation (NTE). This paper investigates the challenges of extending goal-based error estimation to multi-dimensions with supporting evidence provided on 2-D fixed (extraneous) source and Keff eigenvalue (criticality) verification test cases. It was found that extending Hennart’s weighted residual view of the lowest order 1-D DD equations to multi-dimensions gave what has previously been called the box method. This paper shows how the box method can be extended to higher orders. The paper also shows an equivalence between the higher order box methods and the higher order DD methods derived by Hébert et al. Though, less information is retained in the final solution in the latter case. These extensions allow for the definition of dual weighted residual (DWR) error estimators in multi-dimensions for the DD and box methods. However, they are not applied to drive AMR in the multi-dimensional case due to the various challenges explained in this paper.

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.

Duan Y, Cooling C, Ahn JS,
et al., 2019, Using a Gaussian process regression inspired method to measure agreement between the experiment and CFD simulations, *International Journal of Heat and Fluid Flow*, Vol: 80, ISSN: 0142-727X

This paper presents a Gaussian process regression inspired method to measure the agreement between experiment and computational fluid dynamics (CFD) simulation. Because of misalignments between experimental and numerical outputs in spatial or parameter space, experimental data are not always suitable for quantitative assessing the numerical models. In this proposed method, the cross-validated Gaussian process regression (GPR) model, trained based on experimental measurements, is used to mimic the measurements at positions where there are no experimental data. The agreement between an experiment and the simulation is mimicked by the agreement between the simulation and GPR models. The statistically weighted square error is used to provide tangible information for the local agreement. The standardised Euclidean distance is used for assessing the overall agreement.The method is then used to assess the performance of four scale-resolving CFD methods, such as URANS k-ω-SST, SAS-SST, SAS-KE, and IDDES-SST, in simulating a prism bluff-body flow. The local statistically weighted square error together with standardised Euclidean distance provide additional insight, over and above the qualitative graphical comparisons. In this example scenario, the SAS-SST model marginally outperformed the IDDES-SST and better than the other two other, according to the distance to the validated GPR models.

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, 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

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, A VALIDATION OF CFD METHODS ON PREDICTING VALVE PERFORMANCE PARAMETERS, ASME Power Conference 2018 (POWER2018), Publisher: AMER SOC MECHANICAL ENGINEERS

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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

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

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

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

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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.

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