Angular Discretization Research Articles

A boundary-type algorithm based on the discrete ordinates for neutron transport equation

The spatial distribution of neutron flux in the core of a nuclear reactor plays a crucial role in ensuring nuclear safety. It can be determined by solving the neutron transport equation, where computational resource consumption is gradually receiving more attention, especially in problems with multiple reflective boundary conditions. It is common practice to assume initial boundary values in one angular direction to obtain a global solution, and then use reflective boundary conditions to solve for the other directions, iterating until convergence. However, this approach has the drawback of requiring the global solution, and each iteration necessitates storing the neutron flux values, consuming time and storage space. In order to address this issue, this paper proposes a precise and efficient boundary-type method, the Half Boundary Method (HBM). This method establishes relationships between adjacent node values through integration and mathematical deduction, ultimately obtaining the relationship among any node and the boundary values in any direction. As a result, iteration is only required for boundary values, reducing the iteration amount and thus saving time and storage space. Using the discrete ordinate (SN) method for angular discretization and HBM for spatial discretization, this paper solved the steady-state neutron transport equation for two-dimensional systems modeled with Cartesian geometry. The method is validated using several benchmark problems, including the 2D ISSA benchmark, the 2D mono-group k-eigenvalue problem and BWR rod bundle test problem. All of these problems demonstrate good results and effectively reduced computational time.

Numerical investigation of optical characterization of polycarbonate panels

Polycarbonate panels (PC panels) are state-of-the-art transparent insulating materials widely used in the construction industry due to their cavity structure, which provides exceptional thermal insulation and optimal optical performance. However, the inherent anisotropy of the three-dimensional cavity structure complicates radiative transfer and requires consideration of both azimuth and zenith angles in optical performance evaluation. This aspect has received limited attention in existing research. This study aims to accurately characterize the optical performance of PC panels through numerical simulations. A three-dimensional radiative transfer model based on the discrete ordinate radiation model is developed to solve the radiation transfer equation. The model's independency regarding mesh division, angular discretization, and accuracy is validated. The effects of incidence angle, geometric parameters, and optical properties of PC panels on optical performance are analyzed. The findings reveal a strong correlation between transmittance and absorption with variations in incident zenith and azimuth angles. The transmittance exhibits a consistent monotonic variation expressible as a rational bifunction. Notably, absorption peaks occur within specific solid angle ranges, with increased structural complexity resulting in heightened absorption and greater uncertainty. For conventional PC materials, maximum transmittance ranges from 46.9 % to 73 %, while maximum absorption ranges from 2.3 % to 13.5 %. Increasing absorption coefficients, refractive index, and surface scattering coefficients nonlinearly decrease transmittance while increasing absorption. Additionally, deviations in transmittance and absorption with azimuth angle amplify with an increase in non-horizontal structures. Sensitivity analysis indicates a significant influence of zenith angle on transmittance, and absorption coefficient predominantly affects absorption.

A First collision source method using spherical harmonics method for arbitrary shape and order finite element mesh

The discrete ordinates (SN) method is a highly effective and accurate angular discretization method for the neutron transport equation, but suffers from the inherent ray effect in low-scattering regions. The first collision source (FCS) method employing the methods without ray effect to obtain the uncollided flux is a common ray effect mitigation technique. In this paper, a novel FCS method using the spherical harmonics (PN) method to calculate the uncollided flux is proposed. The PN for the first collision source (PN-FCS) method possesses distinctive advantages in several aspects: (1) it can be applied in the problem containing reflective boundary conditions that are difficult to handle with traditional FCS method, (2) it uses the discontinuous Galerkin method to perform spatial discretization, and thus can be easily applied in arbitrary shape and order finite element meshes, (3) it can be effectively calculated in parallel on distributed meshes, (4) the accuracy and computational cost can be flexibly adjusted. Besides, the filter technique is adopted to improve the performance of the PN method in stream-dominated regions. The PN-FCS method has been implemented in two- and three-dimensional neutron transport code and tested by various testing problems. Numerical results show that the PN-FCS method can effectively mitigate the ray effect.

Multi-Objective Robust Beamforming for Integrated Satellite and Aerial Networks Supporting Heterogeneous Services

An integrated satellite and aerial network (ISAN) is considered a promising candidate to provide seamless connectivity for future wireless communication systems. In this paper, we propose a multi-objective based robust beamforming (BF) scheme for an ISAN to support heterogeneous services with high flexibility, where the satellite network serves various heterogeneous satellite terminals through multicast non-orthogonal multiple access (MC-NOMA), while the aerial network offers services to many internet of things devices using layered division multiplexing (LDM). Specifically, we first formulate a multi-objective optimization problem (MOOP) to achieve a good trade-off between sum rate maximization and total transmit power minimization. To tackle this mathematically intractable problem, we exploit the weighted Tchebycheff approach to transform the MOOP into a single-objective problem. Since only the angular information based channel state information is available, we exploit the angular discretization method and sequential convex approximation to design a robust BF algorithm to obtain the Pareto optimal solutions. Finally, simulation results demonstrated that our proposed scheme can achieve a optimal trade-off between multiple performance metrics with high spectrum and energy efficiency, so as to support heterogeneous services in the ISAN and fill the gap of only single type of serivce in the existing ISAN works.

Numerical prediction of fire dynamics and the safety zone in large‐scale multiple pool fire in a dike using flamelet model

AbstractThis research aims to develop a computational fluid dynamics (CFD) methodology for estimating the safety zone around a dike with multiple pool fire (MPF). This study predicts the safety zone and burning characteristics of heptane MPF inside a square dike in calm wind and worst‐case crosswind situations using unsteady simulations. The heptane MPF is modelled using flamelet approach with large eddy simulation (LES) turbulence model incorporating the soot generation. Discrete ordinates and Moss–Brookes model estimate the effect of participating medium radiation on prediction of safety distance and soot production. The discretization is incorporated using an isotropic trimmed cell mesher with local refinement to resolve the flame characteristics in Simcenter STAR CCM+. An extensive grid‐independence study has been executed to find the optimal mesh. The flame temperature and O2 and CO2 mass fraction predictions in calm wind conditions are in good agreement with the experimental findings of Koseki and Yumoto and the maximum flame temperature predictions are within 2.8% error. The safety zone is predicted using the estimated radiative heat flux. The effect of different ordinate sets (angular discretization S4, S6, and S12) approach on safety distance prediction is investigated. The validated grid independent CFD model combined with flamelet generated manifold model and LES turbulence model is proposed to predict the safety zone for industrial MPF in crosswind scenarios, thereby preventing human fatalities and property loss.

Evaluation of angular resolution requirements in the finite-volume-method solution of the radiative transfer equation

The performance of the finite volume method (FVM) used to solve the radiative transfer equation (RTE) is investigated in a simple benchmark configuration corresponding to a hot plate radiating towards sensors located at various separation distances. The study is performed using the Fire Dynamics Simulator (FDS). The objective of this study is to evaluate angular resolution requirements in the FVM-RTE solver to provide suitable accuracy and to avoid ray effects. The FDS-simulated radiative heat fluxes are compared to exact analytical expressions based on view factors. It is found that the numerical error has spatial and angular discretization components. The spatial discretization error is controlled by the size of computational cells and is generally small. The angular discretization error is controlled by the number of radiation angles (NRA) and becomes large in the far-field unless NRA is sufficiently large. A design criterion is proposed to select suitable values of NRA to achieve converged numerical solutions. This criterion is applied to the case of radiant emissions from a benchmark pool fire configuration; this application illustrates how the choice of NRA should be adjusted depending on the size of the region where accurate calculations of radiative heat fluxes are expected.

An Asymptotic-Preserving Hybrid Angular Discretization for the Gray Radiative Transfer Equations

The aim of this paper is to construct a new numerical scheme for the nonlinear gray radiative transfer (GRT) equations, namely, the asymptotic-preserving (AP) H N T -based unified gas kinetic scheme (UGKS). The constructed scheme is obtained by combing the UGKS for spatial discretization with the hybrid H N T method for angular discretization. Since the H N T is a hybrid angular discrete method of both P N and S N methods, the current H N T -based UGKS can not only mitigate the ray effects of the S N method largely, but also suppress the oscillations of the original P N method. Furthermore, we show that the current H N T -based UGKS also inherits the AP property of UGKS. A number of one-dimensional and two-dimensional numerical experiments are presented that validate the performance of the current scheme in both optically thin and thick regimes, as well as in mitigating the ray effects. Moreover, it can capture the initial layer solution without requiring additional treatments.

Angular Adaptivity in P0 Space and Reduced Tolerance Solves for Boltzmann Transport

Previously, we developed an adaptive method in angle that is based on solving in Haar wavelet space with a matrix-free multigrid for Boltzmann transport problems. This method scalably mapped to the underlying P0 space during every matrix-free matrix-vector product; however, the multigrid method itself was not scalable in the streaming limit. To tackle this, we recently built an iterative method based on using an Approximate Ideal Restriction multigrid with GMRES polynomials (AIRG) for Boltzmann transport that showed scalable work with uniform P0 angle in the streaming and scattering limits. This paper details the practical requirements of using this new iterative method with angular adaptivity. Hence, we modify our angular adaptivity to occur directly in P0 space rather than the Haar space. We then develop a modified stabilization term for our Finite Element Method that results in scalable growth in the number of nonzeros in the streaming operator with P0 adaptivity. We can therefore combine the use of this iterative method with P0 angular adaptivity to solve problems in both the scattering and the streaming limits, with close to fixed work and memory use.We also present a coarse-fine splitting for multigrid methods based on element agglomeration combined with angular adaptivity, which can produce a semicoarsening in the streaming limit without access to the matrix entries. The equivalence between our adapted P0 and Haar wavelet spaces also allows us to introduce a robust convergence test for our iterative method when using regular adaptivity. This allows the early termination of the solve in each adapt step, reducing the cost of producing an adapted angular discretization.

Robust Downlink Transmission Design in IRS-Assisted Cognitive Satellite and Terrestrial Networks

Cognitive satellite and terrestrial network (CSTN) is considered as a promising technology to provide ubiquitous connectivity for various users within wide-coverage. This paper proposes a robust downlink transmission scheme for multiple intelligent reflecting surfaces (IRSs) assisted CSTN. Here, the satellite network adopts multigroup multicast transmission scheme to serve many earth stations, while the terrestrial network exploits space division multiple access and multi-IRS-enhanced non-orthogonal multiple access technology to communicate with many terrestrial users. By assuming that these two networks share the same frequency band having only the angular information based imperfect channel state information of each user, we formulate an optimization problem to minimize the total transmit power subject to the constraints of quality-of-service requirement for each user, per-antenna transmit power budgets of satellite and BS, and unit-modulus requirement for each reflecting element. To tackle this mathematically intractable problem, we then employ angular discretization together with the successive convex approximation method to obtain the active beamforming (BF) vectors of satellite and BS, the passive BF vector of IRS, and the power allocation coefficients. Moreover, we propose a generalized zero forcing BF and alternative optimization to obtain the suboptimal solutions of the optimization problem with low computational complexity. Finally, simulation results are given to demonstrate the effectiveness and superiority of the proposed two schemes over the benchmarks.

A finite element method for angular discretization of the radiation transport equation on spherical geodesic grids

Discrete ordinate (SN) and filtered spherical harmonics (FPN) based schemes have been proven to be robust and accurate in solving the Boltzmann transport equation but they have their own strengths and weaknesses in different physical scenarios. We present a new method based on a finite element approach in angle that combines the strengths of both methods and mitigates their disadvantages. The angular variables are specified on a spherical geodesic grid with functions on the sphere being represented using a finite element basis. A positivity-preserving limiting strategy is employed to prevent non-physical values from appearing in the solutions. The resulting method is then compared with both SN and FPN schemes using four test problems and is found to perform well when one of the other methods fail.

Residual distribution schemes for steady radiative transfer equations on unstructured meshes

We consider the numerical solution of steady radiative transfer equations on unstructured meshes. The radiative transfer equations belong to a class of integro-differential equations. We apply conservative residual distribution (RD) methods to solve the radiative transfer equations. To achieve this, we first adopt the discrete ordinate method for angular discretization and use the RD methods to solve the resulting system of coupled linear hyperbolic equations. We present some basics and accuracy analysis for the RD schemes. The standard nonlinear limiter is adopted to upgrade the monotonicity preserving first-order scheme for a high-order one. Several numerical experiments show that the proposed RD schemes achieve second-order accuracy and verify the non-oscillatory property of our numerical schemes.

Improved treatment of multi-material cells in thermal radiation transport codes

High-energy-density physics simulations with non-conformal meshes of materials require a multi-material (MM) closure that affects the thermal radiation transport (TRT). We propose a set of novel closures that work for an arbitrary number of materials, both grey and multigroup energy discretizations, and any angular discretization (such as Sn, IMC, or diffusion). For each spatial cell, our closures let each species (ion and electron) of each material have its own temperature, density, and internal energy, but use a single radiation distribution that interacts with all materials within the cell. Our closures maintain energy conservation, do not incur increased computational cost in the TRT solve itself, are compatible with single-material TRT solvers, and do not make any temperature-equilibrium assumptions. We test our closures on a wide range of increasingly realistic problems and find them to be robust.

A numerical and experimental comparison of heat transfer in a quasi two-dimensional packed bed

Heat transfer plays a major role in many industrial processes taking place in packed beds. An accurate and reliable simulation of the heat exchange between particles is therefore crucial for a reliable operation and to optimize the processes in the bed. The discrete ordinates method (DOM) provides an established numerical technique to model radiative heat transfer in granular media that offers the possibility to consider the directional dependence of the radiation propagation. In this work, DOM is compared with Monte Carlo ray tracing, which provides an alternative method for heat transfer simulations. Geometrically simple configurations are used to investigate the influence of the angular discretization on the accuracy of the results and the computation time in both methods. The obtained insights are then transferred to a more complex configuration of a quasi two-dimensional test rig consisting of metal rods for which also experimental results are available. Our results show that both DOM and Monte Carlo ray tracing allow for an accurate simulation of heat transfer in packed beds. Monte Carlo ray tracing requires thereby computation times that are surprisingly competitive (although still somewhat slower) compared to DOM and allows for an easier computation of highly accurate reference solutions. In our preliminary comparison to the experimental test rig, Monte Carlo ray tracing also provides the advantage that it can more easily model highly specular materials such as stainless steel. Both methods are comparable for diffuse materials such as magnesium oxide.

Radiation Models for Computational Fluid Dynamics Simulations of Photocatalytic Reactors

AbstractThe literature on computational fluid dynamics (CFD) simulations applied to photocatalytic systems is reviewed. CFD simulations referring to three models, namely, P‐1, discrete ordinates (DO), and Monte Carlo (MC) models, for simulating radiation distribution in annular photoreactors are addressed and previous works using these models are reviewed. The cases are a three‐dimensional (3D) annular photoreactor and a two‐dimensional (2D) rectangular enclosure with black walls and the results are compared with the literature. The DO model was selected to solve the radiation transfer equation (RTE) as the most reliable model to fit the radiation distribution inside the reactor. The setting‐up of mesh, angular discretization, and boundary conditions for CFD simulation of photocatalytic reactors are addressed. When correctly set, the CFD models show an excellent agreement with the literature results.

Wavelet-based angular discretization finite difference method for neutron transport equation solving

Angular discretization is key issue for neutron transport equation (NTE) solving because of its integro-differential form, which strongly affects the accuracy of NTE solving. In this paper, the compact supported linear octahedral wavelet is used for angular discretization, and the transformation mapping relationships between sphere and octahedron are presented in detail. The wavelet-based angular discretization NTE is explicitly provided, which can be directly solved with many space–time discrete schemes. The wavelet-based angular discretization FDM (WAFDM) is developed using second-order upwind finite differential method for spatial discretization. Results of several benchmarks demonstrate that the proposed method has good accuracy and stability for neutron transport calculation, and compared with traditional Sn method, the wavelet-based method can obviously improve the ray effect. Besides, the proposed WAFDM is easy to realize wavelet-based angular discretization, and is an effective support for developing angular-space anisotropic and adaptative discretization scheme to solve NTE numerically.

Accelerated Deterministic Phonon Transport With Consistent Material Temperature and Intensities

Abstract We present a method for deterministically solving the frequency and temperature dependent phonon radiative transport (PRT) equation in the single-mode relaxation time (SMRT) approximation in the self-adjoint angular flux (SAAF) form. To handle the nonlinear coupling between the phonon intensities and the material temperature, we apply a linearization approach that is similar to one in thermal radiative transport. This procedure leads to the PRT equation with pseudo-scattering. The method presented includes acceleration of both the inner pseudo-scattering source iterations and outer temperature iteration with a gray diffusion synthetic acceleration (DSA) and Anderson acceleration, respectively. We use the finite-element method to discretize the PRT equation in space and the method of discrete ordinates (SN) for angular discretization. The proposed method is verified by a gray method of manufactured solutions problem and demonstrated on a problem using temperature and direction dependent multigroup data from lithium aluminate (LiAlO2). The iterative performance of the acceleration method in each test is then compared to the unaccelerated method.

Verification of the Discrete Ordinates Goal-Oriented Multi-Collision Source Algorithm with Neutron Streaming Problems

The shielding calculation of neutron streaming problems with ducts is characterized by the strong anisotropy of angular flux, which poses a challenge for the analysis of nuclear installations. The discrete ordinate method is one of the most commonly deterministic techniques to solve the neutron transport equation, in which the accuracy and efficiency neutron are crucial to ensure the reliability of the streaming shielding simulation. We implemented the goal-oriented multi-collision source algorithm in the 3D transport code ARES. This algorithm can determine the importance factor based on the adjoint transport calculation, obtain the response function to enable problem-dependent, goal-oriented spatial decomposition, and provide the error estimation as a driving force behind the dynamic quadrature to optimize the source iteration. This study focuses on verifying the goal-oriented multi-collision source algorithm under the neutron streaming problems, and the capabilities of the algorithm have been tested on IRI-TUB benchmark of SINBAD database. The numerical results show that the algorithm can effectively control the angular discretization error for the neutron streaming problems, which is more economical than the traditional discrete ordinate calculation.

On the simulation of neutron noise induced by vibrations of fuel pins in a fuel assembly

Vibrations of fuel assemblies are an important issue in the safe operation of nuclear reactors, because they can challenge the integrity of the fuel with potential for radioactive releases. Reactor neutron noise-based techniques for monitoring vibrations are valuable for core diagnostic since they are not intrusive and make use of ordinary neutron flux measurements from ex-core and in-core detectors. The application of these techniques involves the solution of inverse problems that require numerical simulations capable of estimating the reactor neutron noise, given a model of the vibrations. For this purpose, several novel reactor neutron noise solvers have been developed in the CORTEX project using either Monte Carlo or deterministic methods, such as the discrete ordinates method, the method of characteristics, and the diffusion approximation. In the current work, these solvers have been scrutinized by computing the neutron noise induced by vibrations of one or multiple fuel pins in a simplified UOX fuel assembly benchmark, via proper variations of macroscopic neutron cross sections. The comparison of these neutron noise solutions obtained from the different methods shows novel insights into the simulation of neutron noise induced by mechanical vibrations, such as the challenges posed by the Monte Carlo method, the impact of the angular discretization on the application of the discrete ordinates method, and the accuracy of the diffusion approximation assessed via the higher-order neutron transport methods.

An hp-angular adaptivity with the discrete ordinates method for Boltzmann transport equation

This paper describes an hp-angular adaptivity algorithm in the discrete ordinates method for Boltzmann transport applications with strong angular effects. This adaptivity uses discontinuous finite element quadrature sets with different degrees, which updates both angular mesh and the degree of the underlying discontinuous finite element basis functions, allowing different angular local refinement to be applied in space. The regular and goal-based error metrics are considered in this algorithm to locate some regions to be refined. A mapping algorithm derived by moment conservation is developed to pass the angular solution between spatial regions with different quadrature sets. The proposed method is applied to some test problems that demonstrate the ability of this hp-angular adaptivity to resolve complex fluxes with relatively few angular unknowns. Results illustrate that a reduction to approximately 1/50 in quadrature ordinates for a given accuracy compared with uniform angular discretization. This method therefore offers a highly efficient angular adaptivity for investigating difficult particle transport problems.

Transient three-dimensional radiative heat transfer analysis of a turbulent diffusion flame-with a focus on the effect of angular discretization scheme

Radiation modeling is a challenging and computational expensive task for turbulent combustion modeling of various fuels. Radiative intensity varies greatly in the angular direction due to turbulent fluctuations of temperature and species fields. A large number of rays are needed in the discrete ordinates method (DOM) or the finite volume method (FVM) to describe the angular variations. Determining the adequate number of rays is essential for the efficient modeling of radiation heat transfer. For this purpose, the effect of angular scheme on predicting the transient three-dimensional radiation heat transfer of a turbulent diffusion flame is investigated. Results show that a smaller number of rays produce strong ray effects and may generate significant errors for radiative heat flux prediction on surrounding walls, while the angular scheme has negligible influence on the radiative heat source. Transient radiative heat fluxes on different walls are also analyzed in detail, and they show completely different contours with unimodal frequency behavior. Recommendation for radiation modeling strategies is proposed for turbulent combustion modeling.