Angular Discretization Research Articles

Numerical investigation of swirl-stabilized pulverized coal flames in air and oxy-fuel atmospheres by means of large eddy simulation coupled with tabulated chemistry

This paper investigates a self-sustained swirl-stabilized pulverized coal combustion chamber, specially designed to validate numerical models under atmospheric and oxy-fuel conditions. For this purpose, a comprehensive model, which accounts for both atmospheric and oxy-fuel conditions, is developed based on Large Eddy Simulation. To describe the turbulent gas-phase chemistry, a 4D Flamelet Generated Manifold based tabulated chemistry model is coupled with the artificially thickened flame approach and implemented in an Euler–Lagrange framework. The chemistry model is able to account for finite rate chemistry, heat losses, mixing devolatilization gases and char off-gases. Radiative heat transfer is included by solving the radiative transfer equation, applying the discrete ordinate method for angular discretization and a weighted sum of gray gas model for the spectral resolution, while particles are treated as gray. The simulation results from the combustion chamber are extensively validated against the available measurement data. Operating conditions with different atmospheres are investigated to validate the models’ prediction accuracy for air and oxy-fuel atmospheres. The validation process is carried out using velocity fields, particle temperature measurements and the gas-species concentrations. Overall, the comparisons between simulations and measurements reveal a favorable agreement with only small differences near the burner opening. These results demonstrate the suitability of large-eddy simulation combined with tabulated chemistry for predicting the combustion of pulverized coal in conventional air and oxy-fuel atmospheres. Further analyses of the self-sustained coal flames are carried out; in particular, the influence of different atmospheres on the interaction of coal particles with the gas phase is studied.

On the Effect of Angular and Spatial Discretization on Perturbation Calculations

In this article, different angular flux discretization options, namely discrete ordinates representation and spherical harmonics expansion are compared from the viewpoint of the accuracy of perturbation calculations. The PARTISN discrete ordinates neutron transport solver was coupled with the SEnTRi code, developed at BME, in order to perform perturbation theory calculations in different types of geometry descriptions and angular representations. With the help of the implemented code, the effect of the angular and spatial discretization on the results of perturbation theory calculations was investigated. Exact matches were observed in Cartesian geometries with the direct perturbation method when the discrete ordinates angular representation was used, and small discrepancies were found when the spherical harmonics expansion was applied. In cylindrical geometries, slight differences were observed with both angular expansions, which originate from the nature of the adjoint transport operator in curvilinear coordinate systems. The differences can be reduced to a negligible level with increased expansion order in both cases. Small discrepancies can have a significant effect in sensitivity, uncertainty and transient calculations, which that for high accuracy calculation the discrete ordinates representation of the angular dependent flux should be used or sufficiently high expansion order with the spherical harmonics must be applied.

Solutions of Chandrasekhar’s basic problem in radiative transfer via theory of functional connections

We present a novel approach to solving Chandrasekhar’s problem in radiative transfer using the recently developed Theory of Functional Connections. The method is designed to elegantly and accurately solve the Linear Boundary Value Problem from the angular discretization of the integrodifferential Boltzmann equation for Radiative Transfer. The proposed algorithm falls under the category of numerical methods for the solution of radiative transfer equations. This new method’s accuracy is tested via comparison with the published benchmarks for Mie and Haze L scattering laws.

A discontinuous and adaptive reduced order model for the angular discretization of the Boltzmann transport equation

SummaryThis article presents a new adaptive reduced order model for resolving the angular direction of the Boltzmann transport equation, based on proper orthogonal decomposition (POD) and the method of snapshots. It builds upon previous methods of applying POD to the angular dimension, with modifications to increase accuracy and solver stability. Previous methods used continuous global functions spanning the whole sphere. The new approach, discontinuous POD (DPOD), partitions the surface of the sphere into angular regions, each with an independent set of POD basis functions. Combined, these can approximate flux distributions which span the sphere using optimized basis functions for each angular region. In addition, a novel implementation of adaptive angular resolution known as adaptive discontinuous POD (ADPOD) is presented, which allows the number of DPOD basis functions to vary by angular octant and spatial element. DPOD and ADPOD are applied to two problems in order to demonstrate their benefits compared with POD. Both are shown to reduce the number of solver iterations required to find a solution and decrease the error in the angular flux.

A comparative study of the delta-Eddington and Galerkin quadrature methods for highly forward scattering of photons in random media

A versatile and accurate treatment for the highly forward-peaked phase function in the three-dimensional (3D) radiative transfer equation (RTE) based on the discrete ordinates method (DOM) is crucial for biomedical optics. Our first objective was to compare the delta-Eddington (dE) and Galerkin quadrature (GQ) methods. The dE method decomposes the phase function into a purely forward-peaked component and the other component, and expands the other component by Legendre polynomials as well as the finite order Legendre expansion (FL) method does. The GQ method conducts the weighting procedure in addition to the Legendre expansion. Although it was reported that both methods can provide the accurate results for calculations of the RTE, the versatility of both methods is still unclear. The second objective was to examine a possibility of a conjunction of the GQ method with the dE method, called as the GQ-dE method, which has the advantages of both methods. We examined numerical errors in the moment conditions of the phase function using the FL, dE, GQ, and GQ-dE methods at various types and orders of the quadrature sets, mainly in the region of the errors induced by the angular discretization using the DOM. The errors were reduced by the dE method from those by the FL method, however the error reduction depended on the types and orders of the quadrature sets. Meanwhile, the errors were significantly reduced by the GQ and GQ-dE methods, regardless of the quadrature sets. We also verified the numerical calculations of the time-dependent 3D RTE by the analytical solution of the RTE for homogeneous media in the region of the scattering length scale, where the highly forward-peaked phase function strongly influences the RTE-results. The errors in the RTE-results were similar to those in the moment conditions. Our results suggest the higher versatility and accuracy of the GQ and GQ-dE methods than those of the FL and dE methods.

Sector sandwich structure: an easy-to-manufacture way towards complex vector beam generation.

Complex polarization-phase transformations that are realized using easy-to-manufacture optical elements are considered. The manufacturing technology of such elements is based on the angular discretization of the required polarization and phase distributions, which allows one to make optical elements in the form of sector sandwich structures consisting of polarized and phase plates stacked together. We analyze analytically and study numerically the main types of such sector sandwich structures for the formation of cylindrical polarizations of various orders. New effects are observed, which result in the appearance of complex polarized beams with vortices of various orders, arising after the passage through polarizing plates and their combinations with differently rotated phase plates. The results of the experimental study of the formed beams using a multichannel diffraction filter are consistent with theory.

Improvements to a class of hybrid methods for radiation transport: Nyström reconstruction and defect correction methods

In this paper, two modifications are introduced for improving the accuracy, versatility, and robustness of a class of hybrid methods for radiation transport. In general, such methods are constructed by splitting the radiative flux into collided and uncollided components to which low- and high-resolution angular approximations are applied, respectively. In this work we focus on discrete ordinates discretizations of high and low order. The first modification we introduce changes the way in which the collided component is mapped into the uncollided component at the end of each time step in a simulation. The new mapping is a Nyström-type reconstruction that is applicable to arbitrary discrete ordinates quadratures, is guaranteed to preserve positivity of the solution provided that all ordinate weights are positive, is significantly more accurate than previous methods, and can be readily extended to other discretizations such as moment methods, finite element methods, and diffusion approximations. The second modification leverages integral deferred correction (IDC) to iteratively correct for the splitting error introduced by the inconsistency in angular discretization between the collided and uncollided components, in addition to improving the accuracy of the low-order temporal error that is treated by traditional IDC methods. Numerical tests in one- and two-dimensional geometries are used to demonstrate the increased accuracy and efficiency of the proposed modifications. It is found that the two techniques combined yield methods with solution accuracy and memory requirements comparable to that of monolithic discrete ordinates methods while reducing runtime by as much as a factor of between two and ten, depending on the problem.

Error analysis and uncertainty quantification for the heterogeneous transport equation in slab geometry

Abstract We present an analysis of multilevel Monte Carlo (MLMC) techniques for the forward problem of uncertainty quantification for the radiative transport equation, when the coefficients (cross-sections) are heterogenous random fields. To do this we first give a new error analysis for the combined spatial and angular discretisation in the deterministic case, with error estimates that are explicit in the coefficients (and allow for very low regularity and jumps). This detailed error analysis is done for the one-dimensional space–one-dimensional angle slab-geometry case with classical diamond differencing. Under reasonable assumptions on the statistics of the coefficients, we then prove an error estimate for the random problem in a suitable Bochner space. Because the problem is not self-adjoint, stability can only be proved under a path-dependent mesh resolution condition. This means that, while the Bochner space error estimate is of order $\mathcal{O}(h^{\eta })$ for some $\eta $ where $h$ is a (deterministically chosen) mesh diameter, smaller mesh sizes might be needed for some realisations. We also show that the expected cost for computing a typical quantity of interest remains of the same order as for a single sample. This leads to rigorous complexity estimates for Monte Carlo (MC) and MLMC: for particular linear solvers, the multilevel version gives up to two orders of magnitude improvement over MC. We provide numerical results supporting the theory.

Goal-based angular adaptivity for Boltzmann transport in the presence of ray-effects

Boltzmann transport problems often involve heavy streaming, where particles propagate long distance due to the dominance of advection over particle interaction. If an insufficiently refined non-rotationally invariant angular discretisation is used, there are areas of the problem where no particles will propagate. These “ray-effects” are problematic for goal-based error metrics with angular adaptivity, as the metrics in the pre-asymptotic region will be zero/incorrect and angular adaptivity will not occur. In this work we use low-order filtered spherical harmonics, which are rotationally invariant and hence not subject to ray-effects, to “bootstrap” our error metric and enable highly refined anisotropic angular adaptivity with a Haar wavelet angular discretisation. We test this on three simple problems with pure streaming in which traditional error metrics fail. We show our method is robust and produces adapted angular discretisations that match results produced by fixed a priori refinement with either reduced runtime or a constant additional cost even with angular refinement.

NECP-hydra: A high-performance parallel SN code for core-analysis and shielding calculation

NECP-Hydra is a three-dimensional high-performance parallel code for reactor core analysis and radiation shielding calculation. This paper summarizes its main features and carries out some verifications. NECP-Hydra empolys finite difference and discrete ordinates for spatial and angular discretization respectively, and it supports both rectangular and cylindrical geometry. The parallelization calculation was performed by the Koch-Baker-Alcouffe (KBA) algorithm based on domain decomposition. Besides existing four-octant sweep (FOS) scheme, a diagonal-octant sweep (DOS) scheme is proposed to further improve the efficiency for the problems with reflective boundaries. Moreover, acceleration methods such as parallel mesh-corner DSA acceleration, inner-group GMRES and multi-group GMRES are also implemented in the code to reduce the computational time. Additional function like continuing computation with surface source for coupling with Monte-Carlo code is also implemented. Finally, we present some numerical results for evaluating NECP-Hydra’s accuracy, efficiency and its preformance in practical analysis of reactors, from which we can draw a conclusion that NECP-Hydra achieves both high accuracy and efficiency throughout these verifications.

Nongray models for radiative absorption and anisotropic scattering by water droplets in fire CFD simulations

An efficient wide band radiation model is developed for absorption and scattering by water droplets in fires. The model applies eight spectral bands that enable coupling with radiating gas and soot media. Optical properties from the Mie theory are averaged within each band and tabulated for six droplet diameters within 10–1000 μm. The table is interpolated/extrapolated as needed during CFD simulations. An energy-conserving, finite-volume angular discretization is also described for efficiently computing anisotropic scattering. These approaches have been implemented in FireFOAM and are demonstrated using the standard emitting-receiving configuration, where a hot and a cold surface are separated by water droplets in transparent gases. The results show that the method conserves energy and predicts centerline incident fluxes within 1% of a previously published, high-fidelity benchmark. The typical normalization approach for energy conservation yields up to 100% error in the fluxes. Validation is performed against a detailed model that combines the Mie theory with a Monte Carlo Ray Tracing (MCRT) solver. Centerline fluxes are predicted within 10% of the detailed model. Gray model predictions are off by up to a factor of two. Finally, the scattering model is shown to introduce minimal overhead (only 15% of radiation solver time).

Effect of Sky Discretization for Shading Device Calculation on Building Energy Performance Simulations

The calculation of sunlit surfaces in a building has always been a relevant aspect in building energy simulation programs. Due to the high computational cost, some programs use algorithms for shading calculation for certain solar positions after discretization of hemispherical sky. The influence of the level of discretization on the estimation of incident direct radiation on building surfaces, as well as on the required computational times, are studied in this work. The direct solar energy on a window for a year, with simulation time steps of five minutes, has been simulated by using an algorithm based on Projection and Clipping Methods. A total of 6144 simulations have been carried out, varying window sizes, window orientations, typologies of shading devices, latitudes and discretization levels of the hemispherical sky. In terms of annual incident solar energy, the results show that maximum error values are about 5% for a low level of angular discretization. Errors up to 22% in hourly incident solar energy have been estimated for some of the configurations analysed. Furthermore, a great number of configurations show errors of shading factor on a window of up to 30%, which could be most relevant in studies of natural lighting. The study also shows that the improvement achieved by the most accurate discretization level implies an increase in computational cost of about 30 times.

A geometry conforming, isogeometric, weighted least squares (WLS) method for the neutron transport equation with discrete ordinate (S[formula omitted]) angular discretisation

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 (SN) 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.

Multi-group lattice Boltzmann method for criticality problems

In this paper, Lattice Boltzmann methodology (LBM) has been developed for the criticality search for the nuclear reactor systems. It is noted that the LBM has not been used for criticality calculations in optically thin (mean free path of neutrons close to system dimensions) and highly anisotropic systems. In order to solve such problems the number of angular directions need to be increased. Various criticality benchmarks with one group and multi-group cross-section including both isotropic and anisotropic scattering have been solved. The solution of these 1D benchmark problems not only verifies the solution methodology, but also demonstrates that multiplication factor of the critical systems having dimension close to mean free path of neutron travel can also be accurately predicted using higher angular discretization. Since LBM approach is relatively simple in implementation it can be effectively used and has a potential to provide easy and convenient solution of neutron transport equation.

A Geometry Conforming, Isogeometric, Weighted Least Squares (WLS) Method for the Neutron Transport Equation with Discrete Ordinate (SN) Angular Discretisation

A Geometry Conforming, Isogeometric, Weighted Least Squares (WLS) Method for the Neutron Transport Equation with Discrete Ordinate (SN) Angular Discretisation

An Adaptive Moving Mesh Discontinuous Galerkin Method for the Radiative Transfer Equation

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, space and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions. Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinate method for angular discretization. The former employs a dynamic mesh adaptation strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency are demonstrated in a selection of one- and two-dimensional numerical examples.

Radiative heat transfer in a 2D semi-transparent gray medium with a centered inner square cavity

Exact semi-analytical expressions of the incident radiation and radiative flux fields are developped, inside a two-dimensional semi-transparent medium and followed by numerical simulations. A rectangular cavity is filled-in with a gray absorbing-emitting, but non-scattering semi-transparent medium at radiative equilibrium. An opaque inner centered cavity with black and imposed temperatures surfaces is set as a non-participating medium, nevertheless its external boudary surfaces participate to radiative transfer. An exact method based on ray tracing and specific functions is performed to evaluate the incident radiation and radiative flux fields. Spatial and angular discretization are ensured with a Gauss quadrature by Numerical issue. Temperature field is computed with an iterative process until convergence. Verification of results is done both for simple and complex geometries, and remains in good agreement with literature. Hence, the other one results of radiative quantities are tabulated as benchmark solutions and simulated for several sizes of the inner square cavity for different ranges of optical thickness.

An asymptotic preserving angular finite element based unified gas kinetic scheme for gray radiative transfer equations

In order to mitigate the ray effects of the usual discrete ordinate (SN) method for the gray radiative transfer equations, a new numerical approach is constructed in this paper with the angular discretization by a linear finite element (FE) method and the spatial discretization by the method of unified gas kinetic scheme (UGKS). Different from the usual SN-based UGKS for which the propagation directions of photons are discretized with finite points, the angular-FE-based UGKS considers the linear combination of all propagation directions (i.e., the unit sphere) and induces a coupling between the discrete directions. Hence, the angular-FE-based UGKS can much mitigate the ray effects to some extent as was expected, as also shown numerically by a point source problem in this paper. At the same time, it is shown that the current scheme possesses the asymptotic preserving (AP) property, that is, in the optically thick regimes the current scheme can exactly capture the solution of the diffusion limit equation without requiring the cell size being smaller than the photon’s mean free path, while the solution in optically thin regimes can also be well resolved in a natural way. Various numerical experiments are included to validate the robustness, accuracy and AP property of the current scheme.

Scalable angular adaptivity for Boltzmann transport

This paper describes an angular adaptivity algorithm for Boltzmann transport applications which for the first time shows evidence of O(n) scaling in both runtime and memory usage, where n is the number of adapted angles. This adaptivity uses Haar wavelets, which perform structured h-adaptivity built on top of a hierarchical P0 FEM discretisation of a 2D angular domain, allowing different anisotropic angular resolution to be applied across space/energy. These wavelets can be mapped back to their underlying P0 space scalably, allowing traditional DG-sweep algorithms if desired. Instead we build a spatial discretisation on unstructured grids designed to use less memory than competing alternatives in general applications and construct a compatible matrix-free multigrid method which can handle our adapted angular discretisation. Fixed angular refinement, along with regular and goal-based error metrics are shown in three example problems taken from neutronics/radiative transfer applications.

Asymptotic convergence of the angular discretization error in the scalar flux computed from the particle transport equation with the method of discrete ordinates

The asymptotic convergence of the angular discretization error in the scalar flux solution of the particle transport equation computed with the Discrete Ordinates (SN) method with increasing quadrature order is examined. Five selected angular quadrature types are considered: Level Symmetric (LS), Legendre-Chebyshev Quadrangular (LCQ), Legendre-Chebyshev Triangular(LCT), Quadruple Range (QR) and Quadruple Range Spence-type (QRS) quadrature sets. We relate the SN angular discretization error to the quadrature error, and split the total flux into the uncollided flux and the fully collided flux, and then we verify the uncollided scalar flux error and the fully collided scalar flux error separately. After developing the theoretical basis for the relationship between true solution regularity and quadrature rule error, we employ a two-dimensional problem to verify the observed and theoretical convergence orders for the region-averaged uncollided and fully collided scalar flux errors. Numerical results show that the angular discretization errors in the region-averaged scalar flux obtained by different quadrature types converge with different rates, that are commensurate with the regularity order of the exact angular flux within the quadrature-designed integration interval. The angular discretization error in the uncollided region-averaged scalar flux obtained by LC class quadratures converges linearly, and the error obtained by QR class quadrature sets converges quadratically. The angular discretization error in the fully collided region-averaged scalar flux converges linearly for LC class quadratures, and faster-than-second order for QR class quadrature sets.