• Nuclear Engineering and Technology
• /
• v.32 no.6
• /
• pp.605-617
• /
• 2000

Nodal transport methods are studied for the solution of two dimensional discrete-ordinates, simplified even-parity transport equation(SEP) which is known to be an approximation to the true transport equation. The polynomial expansion nodal method(PEN) and the analytic function expansion nodal method(AFEN）which have been developed for the diffusion theory are used for the solution of the discrete-ordinates form of SEP equation. Our study shows that while the PEN method in diffusion theory can directly be converted without complication, the AFEN method requires a theoretical modification due to the nonhomogeneous property of the transport equation. The numerical results show that the proposed two methods work well with the SEP transport equation with higher accuracies compared with the conventional finite difference method.

• Nuclear Engineering and Technology
• /
• v.52 no.6
• /
• pp.1137-1147
• /
• 2020

The discrete ordinates method (S_N) is one of the major shielding calculation method, which is suitable for solving deep-penetration transport problems. Our objective is to explore the available quadrature sets and to improve the accuracy in shielding problems involving strong anisotropy. The linear discontinuous finite-element (LDFE) quadrature sets based on the icosahedron (in short, ICLDFE quadrature sets) are developed by defining projected points on the surfaces of the icosahedron. Weights are then introduced in the integration of the discontinuous finite-element basis functions in the relevant angular regions. The multivariate secant method is used to optimize the discrete directions and their corresponding weights. The numerical integration of polynomials in the direction cosines and the Kobayashi benchmark are used to analyze and verify the properties of these new quadrature sets. Results show that the ICLDFE quadrature sets can exactly integrate the zero-order and first-order of the spherical harmonic functions over one-twentieth of the spherical surface. As for the Kobayashi benchmark problem, the maximum relative error between the fifth-order ICLDFE quadrature sets and references is only -0.55%. The ICLDFE quadrature sets provide better integration precision of the spherical harmonic functions in local discrete angle domains and higher accuracy for simple shielding problems.

• Transactions of the Korean Society of Mechanical Engineers
• /
• v.15 no.1
• /
• pp.341-348
• /
• 1991

An efficient tool to deal with a multi-dimensional radiative heat transfer is in strong demand to analyze various thermal problems combined either with other modes of heat transfer or with combustion phenomena. The current study examined the discrete ordinates method (DOM) for a coupled radiative and conductive heat transfer in rectangular enclosures in which either nonscattering or scattering medium is present. The results were compared with the other benchmarked approximate solution. The efficiency and accuracy of the DOM were thus validated.

• Transactions of the Korean Society of Mechanical Engineers B
• /
• v.23 no.1
• /
• pp.104-113
• /
• 1999

A hybrid spatial differencing scheme for the discrete ordinates method is proposed to predict radiative heat transfer in two-dimensional rectangular enclosures. Since this scheme takes the advantages of the diamond scheme and step scheme and includes the characteristics of medium, more accurate and stable results can be obtained. In its development several spatial differencing schemes are examined to address the effect of numerical smearing (or false scattering). Predictions from the proposed hybrid scheme are compared to those of other schemes for transparent, purely absorbing, purely scattering, or absorbing-emitting-isotropically scattering media. It is found that the proposed scheme predicts stable and less smeared results than others.

• Nuclear Engineering and Technology
• /
• v.55 no.2
• /
• pp.769-779
• /
• 2023

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.

• Nuclear Engineering and Technology
• /
• v.34 no.4
• /
• pp.314-322
• /
• 2002

A new extrapolation method is developed and applied to the additive angular dependent rebalance (AADR) acceleration for discrete ordinates neutron transport calculations. With this extrapolation, the convergence of AADR solution for distinct discretizations between the high- order and low-order equations is remarkably improved and thus the “inconsistent discretization problem” is resolved. Fourier analysis is also performed to find the optimal extrapolation and weighting parameters, which give the smallest spectral radius. The numerical tests demonstrate that the AADR with extrapolation works well as predicted by the Fourier analysis.

• Nuclear Engineering and Technology
• /
• v.54 no.7
• /
• pp.2625-2634
• /
• 2022

Discretization errors are extremely challenging conundrums of discrete ordinates calculations for radiation transport problems with void regions. In previous work, we have presented a multi-collision source method (MCS) to overcome discretization errors, but the efficiency needs to be improved. This paper proposes a goal-oriented algorithm for the MCS method to adaptively determine the partitioning of the geometry and dynamically change the angular quadrature in remaining iterations. The importance factor based on the adjoint transport calculation obtains the response function to get a problem-dependent, goal-oriented spatial decomposition. The difference in the scalar fluxes from one high-order quadrature set to a lower one provides the error estimation as a driving force behind the dynamic quadrature. The goal-oriented algorithm allows optimizing by using ray-tracing technology or high-order quadrature sets in the first few iterations and arranging the integration order of the remaining iterations from high to low. The algorithm has been implemented in the 3D transport code ARES and was tested on the Kobayashi benchmarks. The numerical results show a reduction in computation time on these problems for the same desired level of accuracy as compared to the standard ARES code, and it has clear advantages over the traditional MCS method in solving radiation transport problems with reflective boundary conditions.

• Transactions of the Korean Society of Mechanical Engineers B
• /
• v.22 no.2
• /
• pp.213-220
• /
• 1998

The conventional discrete-ordinates method (DOM) is modified and developed for the analysis of two-dimensional axisymmetric cylindrical enclosure with curved wall. The objective of the present work is to extend the capability of the conventional DOM into a general axisymmetric geometry like nozzle-shaped enclosure, by adopting the arbitrary control angle as was done in the finite-volume method (FVM), while keeping the same two-dimensional solution procedure as in the conventional DOM. The present method is validated by applying it to three different benchmark problems of axisymmetric enclosure containing absorbing, emitting and scattering medium. Results presented in this work not only support the solution accuracy, but also moderate efficiency in the numerical calculation of axisymmetric radiation problem.

• Proceedings of the Korean Nuclear Society Conference
• /
• 1996.05a
• /
• pp.173-178
• /
• 1996

특정한 방향성분에 대한 방향중성자속을 정의하는 방향차분 수송 방정식(discrete ordinates or S$_{N}$ transport equation)과 달리 방향변수를 구분된 방향영역에 대하여 적분하고, 해당 방향영역 내에서의 방향중성자속이 일정하다고 가정하는 영역상수법(piecewise constant method)을 이용하여 유사방향차분방정식(discrete ordinates-like equation)을 유도하여, 이를 Boltzmann 수송식과 2계 우성수송식(even-parity transport equation)에 적용하여 기존의 방향차분법의 단점인 광첨두현상(ray effects)을 감소시키고, 우성수송식의 교차미분항을 제거한 단순우성방정식(simplified even-parity equation)을 사용하여 광첨두현상을 제거하였다. 이는 단순우성방정식의 또 다른 장점을 제시한다.

• Korean Journal of Air-Conditioning and Refrigeration Engineering
• /
• v.10 no.1
• /
• pp.95-107
• /
• 1998

Natural convection and radiation heat transfer in a square enclosure containing absorbing, emitting, and isotopically scattering(participating) media is studied numerically using the finite volume method. Various numerical methods are employed to analyze the radiative heat transfer. However, it is very difficult to choose the proper method. In present study, a finite volume method(FVM) and a discrete ordinates method(DOM) are compared in rectangular enclosure. The SIMPLER algorithm is used to solve the momentum and energy equations. Thermal and flow characteristics are investigated according to the variation of radiation parameters such as optical thickness and scattering albedo. The result shows that the accuracy and the computing time of FVM are better than those of DOM in regular geometry.