Adaptive discontinuous finite element quadrature sets over an icosahedron for discrete ordinates method

The discrete ordinates(S N)method requires numerous angular unknowns to achieve the desired accu-racy for shielding calculations involving strong anisotropy.Our objective is to develop an angular adaptive algorithm in the S N method to automatically optimize the angular distribution and minimize angular discretization errors with lower expenses.The proposed method enables linear dis-continuous finite element quadrature sets over an icosahe-dron to vary their quadrature orders in a one-twentieth sphere so that fine resolutions can be applied to the angular domains that are important.An error estimation that operates in conjunction with the spherical harmonics method is developed to determine the locations where more refinement is required.The adaptive quadrature sets are applied to three duct problems,including the Kobayashi benchmarks and the IRI-TUB research reactor,which emphasize the ability of this method to resolve neutron streaming through ducts with voids.The results indicate that the performance of the adaptive method is more effi-cient than that of uniform quadrature sets for duct transport problems.Our adaptive method offers an appropriate placement of angular unknowns to accurately integrate angular fluxes while reducing the computational costs in terms of unknowns and run times.

Convergence of the Modified Discrete Ordinates Method for the Anisotropic Scattering Transport Equation

Criticality problem of nuclear tractors generMly refers to an eigenvalue problem for the transport equations. In this paper, we deal with the eigenvalue of the anisotropic scattering transport equation in slab geometry. We propose a new discrete method which was called modified discrete ordinates method. It is constructed by redeveloping and improving discrete ordinates method in the space of L1（X）. Different from traditional methods, norm convergence of operator approximation is proved theoretically. Furthermore, convergence of eigenvalue approximation and the corresponding error estimation are obtained by analytical tools.

Neutronic calculations of the China dual-functional lithium–lead test blanket module with the parallel discrete ordinates code Hydra

The China dual-functional lithium–lead test blanket module(DFLL-TBM) is a liquid Li Pb blanket concept developed by the Institute of Nuclear Energy Safety Technology of the Chinese Academy of Sciences for testing in ITER to validate relevant tritium breeding and shielding technologies. In this study, neutronic calculations of DFLL-TBM were carried out using a massively parallel three-dimensional transport code, Hydra, with the Fusion Evaluated Nuclear Data Library/MG. Hydra was developed by the Nuclear Engineering Computational Physics Lab based on the discrete ordinates method and has been devoted to neutronic analysis and shielding evaluation for nuclear facilities. An in-house Monte Carlo code(MCX) was employed to verify the discretized calculation model used by Hydra for the DFLL-TBM calculations. The results showed two key aspects:(1) In most material zones,Hydra solutions are in good agreement with the reference MCX results within 1%, and the maximal relative difference of the neutron flux is merely 3%, demonstrating the correctness of the calculation model;(2) while the current DFLL-TBM design meets the operation shielding requirement of ITER for 4 years, it does not satisfy the tritium self-sufficiency requirement. Compared to the two-step approach, Hydra produces higher accuracies as it does not rely on the homogenization technique during the calculation process. The parallel efficiency tests of Hydra using the DFLL-TBM model also showed that this code maintains a high parallel efficiency on O(100) processors and, as a result, is able to significantly improve computing performance through parallelization. Parameter studies have been carried out by varying the thickness of the beryllium armor layer and the tritium breeding zone to understand the influence of the beryllium layer and breeding zone thickness on tritium breeding performance. This establishes a foundation for further improvement in the tritium production performance of DFLL-TBM.

A multithreaded parallel upwind sweep algorithm for the S_(N) transport equations discretized with discontinuous finite elements

The complex structure and strong heterogeneity of advanced nuclear reactor systems pose challenges for high-fidelity neutron-shielding calculations. Unstructured meshes exhibit strong geometric adaptability and can overcome the deficiencies of conventionally structured meshes in complex geometry modeling. A multithreaded parallel upwind sweep algorithm for S_(N) transport was proposed to achieve a more accurate geometric description and improve the computational efficiency. The spatial variables were discretized using the standard discontinuous Galerkin finite-element method. The angular flux transmission between neighboring meshes was handled using an upwind scheme. In addition, a combination of a mesh transport sweep and angular iterations was realized using a multithreaded parallel technique. The algorithm was implemented in the 2D/3D S_(N) transport code ThorSNIPE, and numerical evaluations were conducted using three typical benchmark problems:IAEA, Kobayashi-3i, and VENUS-3. These numerical results indicate that the multithreaded parallel upwind sweep algorithm can achieve high computational efficiency. ThorSNIPE, with a multithreaded parallel upwind sweep algorithm, has good reliability, stability, and high efficiency, making it suitable for complex shielding calculations.

Equivalent low-order angular flux nonlinear finite difference equation of MOC transport calculation

The key issue in accelerating method of characteristics(MOC)transport calculations is in obtaining a completely equivalent low-order neutron transport or diffusion equation.Herein,an equivalent low-order angular flux nonlinear finite difference equation is proposed for MOC transport calculations.This method comprises three essential features:(1)the even parity discrete ordinates method is used to build a low-order angular flux nonlinear finite difference equation,and different boundary condition treatments are proposed;(2)two new defined factors,i.e.,the even parity discontinuity factor and odd parity discontinuity factor,are strictly defined to achieve equivalence between the low-order angular flux nonlinear finite difference method and MOC transport calculation;(3)the energy group and angle are decoupled to construct a symmetric linear system that is much easier to solve.The equivalence of this low-order angular flux nonlinear finite difference equation is analyzed for two-dimensional(2D)pin,2D assembly,and 2D C5G7 benchmark problems.Numerical results demonstrate that a low-order angular flux nonlinear finite difference equation that is completely equivalent to the pin-resolved transport equation is established.

Two Uniform Tailored Finite Point Schemes for the Two Dimensional Discrete Ordinates Transport Equations with Boundary and Interface Layers

This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime,which is valid up to the boundary and interface layers.A five-point nodecentered and a four-point cell-centered tailored finite point schemes(TFPS)are introduced.The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system.Numerically,both methods can not only capture the diffusion limit,but also exhibit uniform convergence in the diffusive regime,even with boundary layers.Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy,uniformly with respect to the mean free path.Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.

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.Geomet-rically 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.

Lattice Boltzmann Method for Conduction and Radiation Heat Transfer in Composite Materials

In this work,the utilization of lattice Boltzmann method(LBM)in the simulation of coupled conduction and radiation heat transfer in composite materials is studied.The novel D3Q30-LBM and D3Q38-LBM models are proposed for the simulation of radiative transfer equation(RTE).The LBM-LBM model,coupled finite volume method(FVM)and LBM are compared with the coupled FVM and discrete ordinate method(DOM)for 2D and 3D simulations.The results show that the original D3Q26-LBM is insufficient for the simulation of radiation,and both the D3Q30-LBM and D3Q38-LBM are close to the DOM for the RTE.The LBM can have large errors in the simulation of heat conduction when the relaxation time is large.Thus,its application in the composite materials is limited when the ratio between thermal conductivities of different components is large.The models with LBM for RTE can be more efficient than the FVM-DOM for the simulation of conduction-radiation heat transfer in composite materials.The FVM-D3Q30-LBM model is suggested because of its accuracy and efficiency.

Preliminary shielding analysis in support of the CSNS target station shutter neutron beam stop design

The construction of China Spallation Neutron Source （CSNS） has been initiated in Dongguan, Guangdong, China. Thus a detailed radiation transport analysis of the shutter neutron beam stop is of vital importance. The analyses are performed using the coupled Monte Carlo and multi-dimensional discrete ordinates method. The target of calculations is to optimize the neutron beamline shielding design to guarantee personal safety and minimize cost. Successful elimination of the primary ray effects via the two-dimensional uncollided flux and the first collision source methodology is also illustrated. Two-dimensional dose distribution is calculated. The dose at the end of the neutron beam line is less than 2.5 μSv/h. The models have ensured that the doses received by the hall staff members are below the standard limit required.

Inverse Radiation Problem of Boundary IncidentRadiation Heat Flux in Semitransparent Planar Slabwith Semitransparent Boundaries

An inverse method is presented for estimating the unknown boundary incident radiation heat flux onone side of one-dimensional semitransparent planar slab with semitransparent boundaries from theknowledge of the radiation intensities exiting from the other side. The inverse problem is solved usingconjugate gradient method of minimization based on discrete ordinates method (DOM) of radiativetransfer equation. The equations of sensitivity coefficients are derived and easily solved by DOM, withthe result that the complicated numerical differentiation commonly used in solving sensitivity coefficients is avoided. The effects of anisotropic scattering, absorption coefficient, scattering coefficient,boundary reflectivity, fluid temperature outside the boundaries, convection heat transfer coefficients,conduction coefficient of semitransparent media and slab thickness on the accuracy of the inverse analysis are investigated. The results show that the boundary incident radiation heat flux can be estimatedaccurately, even with noisy data.

Effects of Flame Temperature and Absorption Coefficient on Premixed Flame Interactions with Radiation

The effects of the flame temperature and the absorption coefficient on interactions between radiative heat transfer and the flame behavior were numerically investigated using high-fidelity numerical code with the discrete ordinates method.To study the effects of the flame temperature and the absorption coefficient,three different flame temperatures and four different absorption coefficient conditions were selected,so twelve test cases were studied in total.In the numerical test results,radiation effects resulted in preheating of the reactant gases and heat loss from the product gases.A higher flame temperature resulted in stronger preheating effects in reactants near the flame.Due to the preheating effects,with the appropriate absorption coefficient,the peak temperature appeared at the flame front.Lower flame temperatures resulted in larger reabsorption effects in the product zone.The peak temperature at the flame front and the flame speed were influenced by the combined effects of preheating and radiative heat loss at the flame front.Depending on the conditions,due to those effects,the peak temperature and the flame speed could increase or decrease.When the absorption coefficient was sufficiently large,the temperature decrease was reduced in the product zone.

An implicit parallel UGKS solver for flows covering various regimes

This paper presents an engineering-oriented UGKS solver package developed in China Aerodynamics Research and Development Center(CARDC).The solver is programmed in Fortran language and uses structured body-fitted mesh,aiming for predicting aerodynamic and aerothermodynamics characteristics in flows covering various regimes on complex three-dimensional configurations.The conservative discrete ordinate method and implicit implementation are incorporated.Meanwhile,a local mesh refinement technique in the velocity space is developed.The parallel strategies include MPI and OpenMP.Test cases include a wedge,a cylinder,a 2D blunt cone,a sphere,and a X38-like vehicle.Good agreements with experimental or DSMC results have been achieved.

Numerical study on the gas-kinetic high-order schemes for solving Boltzmann model equation

The high-order compact finite difference technique is introduced to solve the Boltzmann model equation, and the gas-kinetic high-order schemes are developed to simulate the different kinetic model equations such as the BGK model, the Shakhov model and the Ellipsoidal Statistical (ES) model in this paper. The methods are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the inner flows of normal shock wave for different Mach numbers, and the two-dimensional flows past a circular cylinder and a NACA 002 airfoil to verify the reliability of the present high-order algorithm and simulate gas transport phenomena covering various flow regimes. The computed results are found in good agreement both with the theoretical prediction from continuum to rarefied gas dynamics, the related DSMC solutions, and with the experimental results. The numerical effect of the schemes with the different precision and the different types of Boltzmann collision models on the computational efficiency and computed results is investigated and analyzed. The numerical experience indicates that an approach developing and applying the gas-kinetic high-order algorithm is feasible for directly solving the Boltzmann model equation.

Analysis of Gas Radiative Transfer Using Box Model and Its Comparison with Gray Band Approximation

On the basis of a wide range survey of various models or treatment methods for the calculation of radiative properties of gases, box model, which is similar to the gray band approximation of spectral band model, was applied to evaluate the gas properties in this paper. In order to compare the accuracy of box model with that of gray band approximation of spectral band models, a typical one-dimensional gas radiation problem was analyzed using discrete ordinate method. Comparing with the widely used gray band approximation of narrow band model or exponential wide band model, box model can well evaluate the radiation source term of the radiative problem. It also has the advantages of simplicity and easy to code, so it is practicable and useful for some complex engineering problems.

A Direct Solver for Initial Value Problems of Rarefied Gas Flows of Arbitrary Statistics

An accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space is presented for parallel treatment of rarefied gas flows of particles of three statistics.The discrete ordinate method is first applied to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source term.The high order weighted essentially non-oscillatory scheme is then implemented to capture the time evolution of the discretized velocity distribution function in physical space and time.The method is developed for two space dimensions and implemented on gas particles that obey the Maxwell-Boltzmann,Bose-Einstein and Fermi-Dirac statistics.Computational examples in one-and two-dimensional initial value problems of rarefied gas flows are presented and the results indicating good resolution of the main flow features can be achieved.Flows of wide range of relaxation times and Knudsen numbers covering different flow regimes are computed to validate the robustness of the method.The recovery of quantum statistics to the classical limit is also tested for small fugacity values.

A Gas-Kinetic Unified Algorithm for Non-Equilibrium Polyatomic Gas Flows Covering Various Flow Regimes

In this paper,a gas-kinetic unified algorithm(GKUA)is developed to investigate the non-equilibrium polyatomic gas flows covering various regimes.Based on the ellipsoidal statistical model with rotational energy excitation,the computable modelling equation is presented by unifying expressions on the molecular collision relaxing parameter and the local equilibrium distribution function.By constructing the corresponding conservative discrete velocity ordinate method for this model,the conservative properties during the collision procedure are preserved at the discrete level by the numerical method,decreasing the computational storage and time.Explicit and implicit lower-upper symmetric Gauss-Seidel schemes are constructed to solve the discrete hyperbolic conservation equations directly.Applying the new GKUA,some numerical examples are simulated,including the Sod Riemann problem,homogeneous flow rotational relaxation,normal shock structure,Fourier and Couette flows,supersonic flows past a circular cylinder,and hypersonic flow around a plate placed normally.The results obtained by the analytic,experimental,direct simulation Monte Carlo method,and other measurements in references are compared with the GKUA results,which are in good agreement,demonstrating the high accuracy of the present algorithm.Especially,some polyatomic gas non-equilibrium phenomena are observed and analysed by solving the Boltzmann-type velocity distribution function equation covering various flow regimes.