Quadrature Sets Research Articles

Application of TITAN for Simulation of Particle Streaming in a Duct

The TITAN hybrid deterministic transport code is applied to the simulation of particle streaming in a nuclear power plant duct. A simple model is used consisting of a concrete duct emerging from the pressure vessel with an isotropic surface source with a U-235 fission spectrum located at the pressure vessel end. Multiple methods of simulating the duct using the TITAN code are considered to demonstrate the flexibility of the code and the advantages of TITAN's algorithms. These methods include a discrete ordinates (S N ) calculation, a characteristics method calculation, and the use of a fictitious quadrature set with simplified ray-tracing. The TITAN code's results are compared with MCNP5 solutions. While all TITAN solutions are obtained in a shorter computation time than the MCNP5 solution, the TITAN solution with the fictitious quadrature set shows the largest speedup.

Corner neutronic code

The neutronic code CORNER is based on the SN discrete ordinates method [1] and the PM scattering cross-section approximation. It is intended for high-precision deterministic neutronic calculations of fast-neutron reactors and can be used to solve two types of steady-state neutron transport and gamma quanta problems in a 3D hexagonal geometry: Keff problems (homogeneous) and source problems (inhomogeneous).The code is developed in the Fortran language and has a modular structure. Its key modules are a module for the preparation of neutron constants in the ANISN format; a geometric module containing a description of the core's loading map and fuel assembly types, including their axial meshing and material composition; a module for preparing angular quadrature sets; an input data module containing the parameters of the approximation used and the control parameters; a neutronic calculation module and a calculation data processing module.The Directional Theta-Weighted (DTW) difference scheme [2] has been built to approximate the spatial dependence. It has advantages over the DD (Diamond Difference) scheme broadly used in coarse-mesh problems.The energy dependence is represented by multigroup approximation. The angular variable is discretized by introduction of the angular quadrature set. Quadrature sets can be also defined by the user.An iterative solution process is used, including external iterations for the fission source and internal iterations for the scattering source. The paper presents the results of a cross-verification against the Monte Carlo MMK code [3] and on a model of the BN-800 reactor core.

Lagrange Discrete Ordinates: A New Angular Discretization for the Three-Dimensional Linear Boltzmann Equation

The classical Sn equations of Carlson and Lee have been a mainstay in multidimensional radiation transport calculations. In this paper, an alternative to the Sn equations, the “Lagrange Discrete Ordinates” (LDO) equations, are derived. These equations are based on an interpolatory framework for functions on the unit sphere in three dimensions. While the LDO equations retain the formal structure of the classical Sn equations, they have a number of important differences. The LDO equations naturally allow the angular flux to be evaluated in directions other than those found in the quadrature set. To calculate the scattering source in the LDO equations, no spherical harmonic moments are needed—only values of the angular flux. Moreover, the LDO scattering source preserves the eigenstructure of the continuous scattering operator. The formal similarity of the LDO equations with the Sn equations should allow easy modification of mature three-dimensional Sn codes such as PARTISN or PENTRAN to solve the LDO equations. Numerical results are shown that demonstrate the spectral convergence (in angle) of the LDO equations for smooth solutions and the ability to mitigate ray effects by increasing the angular resolution of the LDO equations.

SU-E-T-270: Diffusion Synthetic Acceleration for Linear Boltzmann Transport Equation

Purpose: Linear Boltzmann transport equation (LBTE) is as accurate as the Monte Carlo method (MC) for dose calculation in photon/particle therapy (LBTE is a deterministic and Eulerian formulation and MC is a statistical and Lagrangian description). An advantage of LBTE is that numerous acceleration techniques can be utilized for acceleration. This work is to explore the acceleration of LBTE via diffusion synthetic acceleration (DSA). Methods: For simplicity, two-dimensional, steady-state, and within-group LBTE is considered with two angular dimensions and two spatial dimensions. The discrete ordinate method is developed for solving this integro-differential equation. The angular variables are discretized using a level-symmetric quadrature set on the unit sphere. The spatial variables are discretized on the structured grid based on the diamond scheme. The source-iteration method (SI) is used to solve the discretized system.Since SI is slow in optically thick and highly scattering regime. DSA is developed to accelerate SI. The motivation for DSA is that diffusion equation (DE) is a good approximation of LBTE in the above regime. However, DE is much cheaper than LBTE computationally since DE only involves spatial variables. Thus, in each DSA iteration, DSA adds to the SI step a computationally-negligible DE step, i.e., to first solve DE with the SI residual as source term, and then compensate the SI solution with DE solution. Results: DSA was benchmarked and compared with SI. The difference between two methods was within 0.12% which verifies the accuracy of DSA, while DSA demonstrated the great advantage in speed, e.g., the reduction of iteration number to 6% and 4% respectively for cases with 100 and 1,000 scattering-absorption ratio that commonly occur in clinical dose calculation. Conclusion: DSA has been developed as one of many possible means for accelerating the numerical solver of LBTE for dose calculation. The authors were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang Talent Program (#14PJ1404500).

Optimized tracking strategies for step MOC calculations in extruded 3D axial geometries

Specialized tracking techniques have been developed for a step-characteristics transport solver. These techniques strongly exploit the regularities of extruded 3D axial geometries to reduce the number of operations and the memory requirements of the transport solver. The method of compound trajectories has been applied to exactly treat geometrical boundary conditions with a minimum number of trajectories. The chord classification method consists in identifying recognized types of chords that share the same length. It allows a consistent reduction of the number of chords that have to be stored and also an important speed up of the transport iterations thanks to the pre-computed values of escape coefficients. An alternative storage method has also been implemented: in this method one trajectory is represented by a partial record which is used to reconstruct the trajectories on-the-fly. Numerical tests show good improvements for both memory and computational time. Optimized methods have also been developed for horizontal and vertical directions, which are particularly advantageous in terms of computational costs. In the same contest, a modified version of the Minimized-Bickley quadrature (Sanchez et al., 2002) containing these directions is proposed. The new methods have been tested by comparison with standard quadrature sets. Results show that for some cases the special directions improve the efficiency of the solver, while for some others the opposite effect is obtained.

The generation of arbitrary order, non-classical, Gauss-type quadrature for transport applications

A method is presented, based upon the Stieltjes method (1884), for the determination of non-classical Gauss-type quadrature rules, and the associated sets of abscissae and weights. The method is then used to generate a number of quadrature sets, to arbitrary order, which are primarily aimed at deterministic transport calculations. The quadrature rules and sets detailed include arbitrary order reproductions of those presented by Abu-Shumays in [4,8] (known as the QR sets, but labelled QRA here), in addition to a number of new rules and associated sets; these are generated in a similar way, and we label them the QRS quadrature sets.The method presented here shifts the inherent difficulty (encountered by Abu-Shumays) associated with solving the non-linear moment equations, particular to the required quadrature rule, to one of the determination of non-classical weight functions and the subsequent calculation of various associated inner products. Once a quadrature rule has been written in a standard form, with an associated weight function having been identified, the calculation of the required inner products is achieved using specific variable transformations, in addition to the use of rapid, highly accurate quadrature suited to this purpose. The associated non-classical Gauss quadrature sets can then be determined, and this can be done to any order very rapidly. In this paper, instead of listing weights and abscissae for the different quadrature sets detailed (of which there are a number), the MATLAB code written to generate them is included as Appendix D.The accuracy and efficacy (in a transport setting) of the quadrature sets presented is not tested in this paper (although the accuracy of the QRA quadrature sets has been studied in [12,13]), but comparisons to tabulated results listed in [8] are made. When comparisons are made with one of the azimuthal QRA sets detailed in [8], the inherent difficulty in the method of generation, used there, becomes apparent, with the highest order tabulated sets showing unexpected anomalies.Although not in an actual transport setting, the accuracy of the sets presented here is assessed to some extent, by using them to approximate integrals (over an octant of the unit sphere) of various high order spherical harmonics. When this is done, errors in the tabulated QRA sets present themselves at the highest tabulated orders, whilst combinations of the new QRS quadrature sets offer some improvements in accuracy over the original QRA sets.Finally, in order to offer a quick, visual understanding of the various quadrature sets presented, when combined to give product sets for the purposes of integrating functions confined to the surface of a sphere, three-dimensional representations of points located on an octant of the unit sphere (as in [8,12]) are shown.

Advanced quadratures for three-dimensional discrete ordinate transport simulations: A comparative study

Three-dimensional radiation transport simulations play a key role in a number of problems found in nuclear science and engineering. When considering deterministic methods for solving the three-dimensional radiation transport equation, the Sn method of Carlson and Lee is one of the most commonly found techniques. The Sn method relies on a given set of “discrete ordinates” or discrete directions in which radiation streams. This set of discrete ordinates also must form a quadrature (numerical integration) so that the angular flux moments can be estimated and the scattering source calculated. In this paper we review a number of recently developed angular quadrature sets and compare their performance on a number of reactor physics problems.

Ray Effects Elimination in Discrete Ordinates and Finite Volume Methods

Ray effects through two- and three-dimensional isotropically scattering, participating media are investigated using four ordinates methods: discrete ordinates, using a classical quadrature scheme and a refined azimuthal discretization around the azimuthal angle endpoints; and finite volume, with and without refined azimuthal discretization around the azimuthal angle endpoints. The discrete ordinates method with refined azimuthal angular discretization consists of a coupling of Gaussian quadrature for polar discretization and control azimuthal angle discretization splitting into a quadruple range. For both the discrete ordinates method with refined azimuthal angular discretization and the finite volume method with refined azimuthal angular discretization, the azimuthal direction is discretized such that azimuthal grid points are close to the azimuthal angle endpoints. Radiative heat transfer through isotropically scattering, participating and optically thin media within enclosures considering uniform and strip diffuse boundary conditions is considered. Excellent agreement between the discrete ordinates method/ finite volume method with refined azimuthal angular discretization and the benchmark literature results is achieved for high-order azimuthal discretization, whereas finer high-order azimuthal discretization is required for the finite volume method. Ray effects are eliminated for combinations of higher-order azimuthal discretization and refined azimuthal discretization around the azimuthal angle endpoints. It has also been found that, in order to eliminate ray effects, the discrete ordinates and finite volume methods’ quadrature sets’ order must be increased as the medium’s aspect ratio decreases or when the size of the diffuse strip of incident radiation on one of the medium’s surfaces is decreased.

General Expressions for Auxiliary Equations of a Nodal Formulation for Two-Dimensional Transport Calculations

A nodal approach is proposed for the solution of discrete ordinates two-dimensional fixed-source problems in Cartesian geometry. For the one-dimensional equations generated by the integration process, explicit solutions in terms of the spatial variables are derived by the analytical discrete ordinates method. The procedure does not require either iterative schemes or domain division into cells, and when associated with the level-symmetric quadrature set it leads to reduced order eigensystems. In particular, the unknown angular fluxes on the boundaries are written in terms of the elementary solutions of the one-dimensional integrated problems and a coupled system for the x and y directions is then obtained. Numerical results generated by the technique are provided and compared to those present in the literature.

WE-A-17A-07: Evaluation of a Grid-Based Boltzmann Solver for Nuclear Medicine Voxel-Based Dose Calculations

Purpose: Grid-based Boltzmann solvers (GBBS) have been successfully implemented in radiation oncology clinics as dose calculations for e×ternal photon beams and 192Ir sealed-source brachytherapy. We report on the evaluation of a GBBS for nuclear medicine vo×el-based absorbed doses. Methods: Vo×el-S-values were calculated for monoenergetic betas and photons (1, 0.1, 0.01 MeV), 90Y, and 131I for 3 mm vo×el sizes using Monte Carlo (DOS×YZnrc) and GBBS (Attila 8.1-beta5, Transpire). The source distribution was uniform throughout a single vo×el. The material was an infinite 1.04 g/cc soft tissue slab. To e×plore convergence properties of the GBBS 3 tetrahedral meshes, 3 energy group structures, 3 different square Chebyschev-Legendre quadrature set orders (Sn), and 4×2013;7 spherical harmonic e×pansion terms (Pn) were investigated for a total of 168 discretizations per source. The mesh, energy group, and quadrature sets are 8×, 3×, and 16×, respectively, finer than the corresponding coarse discretization. GBBS cross sections were generated with full electronphoton-coupling using the vendors e×tended CEP×S code. For accuracy, percent differences (%Δ) in source vo×el absorbed doses between MC and GBBS are reported for the coarsest and finest discretization. For convergence, ratios of the two finest discretization solutions are reported along each variable. Results: For 1 MeV, 0.1 MeV, 0.01 MeV, Y90, and I-131 beta sources the %Δ in the source vo×el for (coarsest,finest) discretization were (+2.0,−6.4), (−8.0, −7.5), (−13.8, −13.4), (+0.9,−5.5), and (− 10.1,−9.0) respectively. The corresponding %Δ for photons were (+33.7,−7.1), (−9.4, −9.8), (−17.4, −15.2), and (−1.7,−7.7), respectively. For betas, the convergence ratio of mesh, energy, Sn, and Pn ranged from 0.991–1.000. For gammas, the convergence ratio of mesh, Sn, and Pn ranged from 0.998–1.003 while the ratio for energy ranged from 0.964–1.001. Conclusions: GBBS is promising for nuclear medicine vo×el-based dose calculations. Ongoing work includes evaluating GBBS in bone, lung, and realistic clinical PET/SPECT-based activity distributions. Research reported in this publication was supported by the National Cancer Institute of the National Institutes of Health under Award Number R01CA138986. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

Unique formulations in TITAN and PENTRAN for medical physics applications

Special algorithms for application to medical physical problems have been developed and implemented into the TITAN and PENTRAN 3-D parallel transport codes. TITAN is a 3-D parallel transport code with hybrid algorithms including 1) Sn and Characteristics, and 2) Sn with a fictitious quadrature set and ray-tracing. PENTRAN is a 3-D parallel transport code with adaptive differencing Sn formulation, full domain decomposition, and different angular quadrature types (including a characteristic option) with ordinate splitting. In this paper, we will discuss new algorithms developed in TITAN for image reconstruction in Single Photon Emission Computed Tomography (SPECT). It is demonstrated that TITAN projection images are in excellent agreement with the Monte Carlo MCNP5 predictions, while its computation time was up to 2787 times faster than the MCNP5 computation time for a 0.98° collimator acceptance angle. We will also discuss a novel algorithm developed in PENTRAN for dose calculations in heterogeneous, voxelized phantoms or other geometries. The new algorithm called EDK-Sn, or “Electron Dose Kernel-Discrete Ordinates (EDK-Sn)” yielded a speedup of ~8 over traditional highly parallel Monte Carlo calculation times, with <7% difference in dose among different organs (or smaller given stochastic uncertainties).

Monte Carlo and Deterministic Neutronics Analyses of YALINA Thermal Facility and Comparison with Experimental Results

This study compares Monte Carlo and deterministic neutronics analyses of the zero-power YALINA Thermal subcritical assembly, which is located in Minsk, Belarus. The YALINA Thermal facility consists of a subcritical core that can be driven by either a californium neutron source or a deuterium-deuterium (D-D) neutron source. The californium neutron source is generated by the natural decay of 252Cf; the D-D neutron source is generated by a deuteron accelerator. The MCNPX, MONK, NJOY, DRAGON, PARTISN, and TORT computer programs have been used for calculating the neutron spectrum, the neutron flux, and the 3He(n,p) reaction rate set by californium and D-D neutron sources. These parameters have been computed in different experimental channels of the assembly for different fuel loading configurations. The MCNPX and MONK computer programs modeled the facility without any major approximation; the PARTISN and TORT computer simulations used 69 energy groups, S16 angular quadrature set, linear anisotropic scattering, and approximately 60 homogenized material zones. The results calculated by different computer programs are in good agreement; in addition, they match the 3He(n,p) reaction rate from experimental measurements obtained by californium and D-D neutron sources.

Prompt Fission Neutron Spectrum Uncertainty Propagation Using Polynomial Chaos Expansion

The polynomial chaos expansion-stochastic collocation method (PCE-SCM) is demonstrated to be a computationally efficient approach for propagating nuclear data uncertainties evaluated for the prompt fission neutron spectra (PFNS) of n + 235U and n + 239Pu fission reactions through two fast neutron critical benchmark experiments. A principal component decomposition of the PFNS covariance matrices yields an efficient representation of the uncertainty in terms of two to four random variables. Both normal and uniform distributions are considered for these random variables, and the random output variables (angular flux and k-eigenvalue) are expressed in terms of Hermite and Legendre chaos expansions, respectively. Tensor product Hermite and Legendre Gauss quadrature sets, respectively, are used to relate the deterministic chaos expansion coefficients to solutions of independent transport k-eigenvalue problems, and the resulting polynomial chaos expansion provides a complete statistical characterization of the uncertainty in the output variables. Direct random sampling of the PFNS followed by repeated solution of the transport problem to create an ensemble of solutions is used to benchmark results obtained from the PCE-SCM implementation. Both direct random sampling and the PCE-SCM implementation yield comparable results where, for the Jezebel and Lady Godiva critical assemblies, the calculated uncertainties in keff resulting from the PFNS propagated uncertainties are found to be of the same order or larger than reported experimental measurement uncertainties, respectively. The PCE-SCM implementation results obtained require orders of magnitude less computational resources compared with the direct random sampling approach.

Summary of the Up-to-Date 3-D Nuclear Analyses of ITER Diagnostics Generic Equatorial Port Plug (GEPP) Performed with the Attila Design Code

The upper, equatorial, and lower diagnostics port plugs in ITER will include numerous intermingling labyrinths and many streaming paths whose impact should be carefully studied. For this purpose, the 3-D Discrete Ordinates code, Attila, has been routinely used by PPPL/UCLA to assess the nuclear field in these geometrically complex plugs both during operation and after shutdown. In this paper we describe the calculation procedure followed and the input parameters/assumptions applied to assess the shutdown dose rates (SDDR) everywhere with emphasize on their values inside the generic equatorial port plug (GEPP) and its inter-space extension area. Factors inherent in the Discrete Ordinates method that impact the accuracy of the results (e.g. quadrature sets used, boundary conditions applied, etc.) are discussed. Means to minimize streaming through straight gaps and open channels present in the GEPP are presented in this paper, along with an examination of their effectiveness in reducing the SDDR in the port inter-space area.

THERMAL INVESTIGATION OF COMMON 2D FETs AND NEW GENERATION OF 3D FETs USING BOLTZMANN TRANSPORT EQUATION IN NANOSCALE

The thermal performance of two-dimensional (2D) field-effect transistors (FET) is investigated frequently by solving the Fourier heat diffusion law and the Boltzmann transport equation (BTE). With the introduction of the new generation of 3D FETs in which their thickness is less than the phonon mean-free-path it is necessary to carefully simulate the thermal performance of such devices. This paper numerically integrates the BTE in common 2D transistors including planar single layer and Silicon-On-Insulator (SOI) transistor, and the new generation of 3D transistors including FinFET and Tri-Gate devices. In order to decrease the directional dependency of results in 3D simulations; the Legendre equal-weight (PN-EW) quadrature set has been employed. It is found that if similar switching time is assumed for 3D and 2D FETs while the new generation of 3D FETs has less net energy consumption, they have higher hot-spot temperature. The results show continuous heat flux distribution normal to the silicon/oxide interface while the temperature jump is seen at the interface in double layer transistors.

Comparison of Quadrature Schemes in DOM for Anisotropic Scattering Radiative Transfer Analysis

The commonly implemented level-symmetric S N quadrature set for the discrete-ordinates method suffers from a limitation in discrete direction number to avoid physically unrealistic weighting factors. This limitation can have an adverse impact for determining radiative transfer, as directional discretization results in angular false scattering errors due to distortion of the scattering phase function in addition to the ray effect. To combat this limitation, several higher-order quadrature schemes with no directional limitation have been developed. Here, four higher-order quadrature sets (Legendre-equal weight, Legendre-Chebyshev, triangle tessellation, and spherical ring approximation) are implemented for determination of radiative transfer in a 3-D cubic enclosure containing participating media. Heat fluxes obtained at low direction number are compared to the S N quadrature and Monte Carlo predictions to gauge and compare quadrature accuracy. Investigation into the reduction/elimination of angular false scattering with increase in direction number, including heat flux accuracy with respect to Monte Carlo and computational efficiency, is presented. It is found that while the higher-order quadrature sets are able to effectively minimize angular false scattering, the number of directions required is extremely large, and thus it is more computationally efficient to implement proper phase-function normalization to obtain accurate results.

Analytical methods for computational modeling of fixed-source slab-geometry discrete ordinates transport problems: Response matrix and hybrid SN

Presented here are two analytical methods for computational modeling of fixed-source slab-geometry discrete ordinates (SN) transport problems for shielding and nuclear reactor fuel-moderator lattice calculations. For shielding calculations a response matrix method is described, which generates numerical solutions completely free from spatial truncation errors. Therefore, the offered response matrix method with the one-region block inversion (RBI) iterative scheme converges numerical results for the region-edge angular fluxes, that coincide with the numerical values generated from the SN analytical solution, apart from computational finite arithmetic considerations. As with lattice cell calculations in nuclear reactor physics, we describe an analytical direct method for hybrid SN calculations. The basic idea is to use higher order angular quadrature set in the highly absorbing fuel region (SNF) and lower order angular quadrature set in the diffusive moderator region (SNM), i.e., NF > NM. Special continuity conditions for the fuel exiting fluxes that constitute the incoming fluxes for the moderation region, and conversely, for the moderator exiting fluxes that constitute the incoming fluxes for the fuel region are applied in the present hybrid numerical scheme. These special continuity conditions are based on the equivalence of the multigroup SN and PN − 1 equations in slab geometry. A spatial reconstruction scheme is added to yield detailed profile of the solution within each homogenized region of the slab, starting from the coarse-mesh results. Test problems are given to illustrate the methods' accuracy.

Asymptotic P N -Equivalent S N+1 Equations

The 1-D one-speed slab-geometry P N equations with isotropic scattering can be modified via an alternative moment closure to preserve the two asymptotic eigenmodes associated with the transport equation. Pomraning referred to these equations as the asymptotic P N equations. It is well-known that the 1-D slab-geometry S N+1 equations with Gauss quadrature are equivalent to the standard P N equations. In this article, we first show that if any quadrature set meets a certain criterion, the corresponding S N+1 equations will be equivalent to a set of P N equations with a quadrature-dependent closure. We then derive a particular family of quadrature sets that make the S N+1 equations equivalent to the asymptotic P N equations. Next we theoretically demonstrate several of the properties of these sets, relate them to an existing family of quadratures, numerically generate several example quadrature sets, and give numerical results that confirm several of their theoretically predicted properties.

Phase-Function Normalization in the 3-D Discrete-Ordinates Solution of Radiative Transfer—PART I: Conservation of Scattered Energy and Asymmetry Factor

The conditions for which conversation of scattered energy and phase-function asymmetry factor after discrete-ordinates methods (DOM) directional discretization for 3-D radiative transfer in anisotropic scattering media breaks down are examined. Directional discretization in anisotropic scattering media is found to alter the scattering asymmetry factor—a second-type of “false scattering.” Phase-function normalization which conserves scattered energy alone cannot correct this problem, and conservation of the asymmetry factor is simultaneously required. A normalization technique developed by the authors, which was successfully tested in 2-D asymmetric cylindrical-coordinate radiative transfer analysis, is intensively examined and validated with benchmark problems in 3-D Cartesian coordinates. In Part I of this study, the degree of anisotropy for which normalization is necessary to conserve these inherent quantities is presented for various phase-function approximations and discrete quadrature sets.

A Comparison of Angular Discretization Strategies for Modeling Radiative Transfer in Pool Fire Simulations

Decoupled radiative heat transfer calculations of 30-cm-diameter heptane pool fire are carried out employing the P1, discrete ordinates (DO), and finite-volume (FV) methods. The DO calculations are performed employing three different quadrature schemes (level symmetric hybrid [LSH], spherical surface symmetrical equal dividing angular quadrature [SSD], and TN). By scaling the soot field in the fire, the performances of the various methods are evaluated at different optical thicknesses by employing highly angularly resolved FV calculations (512 directions) as a reference. Twenty-four angular directions in the DO and FV methods are seen to be adequate to accurately predict the distributions of the divergence of the radiative flux within the fire. The corresponding accuracies of the P1 model are seen to decrease with increase in optical thickness. In the regions away from the fire, the P1 model performs poorly in predicting the axial as well as the radial radiative fluxes, whereas the errors associated with employing coarse angular resolutions in the DO method are seen to increase with soot concentrations and distance from the pool centre. Among the DO and FV models, the accuracies of the axial and radial incident radiative flux predictions outside the fire were more dependent on the angular resolution of the calculations rather than on the particular quadrature set employed in the calculations.