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

• Journal of Energy Engineering
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• v.16 no.1 s.49
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• pp.46-52
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• 2007

The piecewise constant angular approximation is developed to replace the conventional angular quadrature sets in the solution of the second-order, multi-dimensional $S_{N}$ neutron transport equations. The newly generated quadrature sets by this method substantially mitigate ray effects and can be used in the same manner as the conventional quadrature sets are used. The discrete-ordinates and the piecewise-constant approximations are applied to both the first-order Boltzmann and the second-order form of neutron transport equations in treating angular variables. The result is that the mitigation of ray effects is only achieved by the piecewise-constant method, in which new angular quadratures are generated by integrating angle variables over the specified region. In other sense, the newly generated angular quadratures turn out to decrease the contribution of mixed-derivative terms in the even-parity equation that is one of the second-order neutron transport equation. This result can be interpreted as the entire elimination or substantial mitigation of ray effect are possible in the simplified even-parity equation which has no mixed-derivative terms.

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

• Nuclear Engineering and Technology
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• v.49 no.6
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• pp.1114-1124
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• 2017

Acceleration/preconditioning strategies available in the SCEPTRE radiation transport code are described. A flexible transport synthetic acceleration (TSA) algorithm that uses a low-order discrete-ordinates ($S_N$) or spherical-harmonics ($P_N$) solve to accelerate convergence of a high-order $S_N$ source-iteration (SI) solve is described. Convergence of the low-order solves can be further accelerated by applying off-the-shelf incomplete-factorization or algebraic-multigrid methods. Also available is an algorithm that uses a generalized minimum residual (GMRES) iterative method rather than SI for convergence, using a parallel sweep-based solver to build up a Krylov subspace. TSA has been applied as a preconditioner to accelerate the convergence of the GMRES iterations. The methods are applied to several problems involving electron transport and problems with artificial cross sections with large scattering ratios. These methods were compared and evaluated by considering material discontinuities and scattering anisotropy. Observed accelerations obtained are highly problem dependent, but speedup factors around 10 have been observed in typical applications.

• Nuclear Engineering and Technology
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• v.28 no.5
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• pp.488-499
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• 1996

We have extended the source projection analytic nodal discrete ordinates method (SPANDOM) for more flexible applicability in analysis of hexagonal assembly cores. The method (SPANDOM-FH) does not invoke transverse integration but instead solves the discrete ordinates equation analytically after the source term is projected and represented in hybrid form of high-order polynomials and exponential functions. SPANDOM-FH which treats a hexagonal node as one node is applied to two fast reactor benchmark problems and compared with TWOHEX. The results of comparison indicate that the present method SPANDOM-FH predicts accurately $k_eff$ and flux distributions in hexagonal assembly cores. In addition, SPANDOM-FH gives the continuous two dimensional intranodal scalar flux distributions in a hexagonal node. The reentering models between TWOHEX and SPANDOM were also compared and it was confirmed that SPANDOM's model is more realistic. Through the results of benchmark problems, we conclude that SPANDOM-FH has the sufficient accuracy for the nuclear design of fast breeder reactor (FBR) cores with hexagonal assemblies.

• Nuclear Engineering and Technology
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• v.55 no.2
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• pp.756-768
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• 2023

Neutron transport calculations are extremely challenging due to the high computational cost of large and complex problems. A multilevel octree grid algorithm (MLTG) of discrete ordinates method was developed to improve the modeling accuracy and simulation efficiency on 3-D Cartesian grids. The Balakovo-3 VVER-1000 neutron dosimetry benchmark is calculated to verify and validate this numerical technique. A simplified S₂ synthetic acceleration is used in the MLTG calculation method to improve the convergence of the source iterations. For the triangularly arranged fuel pins, we adopt a source projection algorithm to generate pin-by-pin source distributions of hexagonal assemblies. MLTG provides accurate geometric modeling and flexible fixed source description at a lower cost than traditional Cartesian grids. The total number of meshes is reduced to 1.9 million from the initial 9.5 million for the Balakovo-3 model. The numerical comparisons show that the MLTG results are in satisfactory agreement with the conventional S_N method and experimental data, within the root-mean-square errors of about 4% and 10%, respectively. Compared to uniform fine meshing, approximately 70% of the computational cost can be saved using the MLTG algorithm for the Balakovo-3 computational model.

• Transactions of the Korean Society of Mechanical Engineers
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• v.19 no.11
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• pp.2995-3002
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• 1995

The cooling problem of the hot internal pipe flow has been investigated. Simultaneous conduction, convection, and radiation were considered with azimuthally varying convective heat loss at the pipe wall. A complex, nonlinear integro-differential radiative transfer equation was solved by the discrete ordinates method (or called S$_{N}$ method). The energy equation was solved by control volume based finite difference technique. A parametric study was performed by varying the conduction-to-radiation parameter, optical thickness, and scattering albedo. The results have shown that initially the radiatively active medium could be more efficiently cooled down compared with the cases otherwise. But even for the case with dominant radiation, as the medium temperature was lowered, the contribution of conduction became to exceed that of radiation.n.

• Nuclear Engineering and Technology
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• v.54 no.11
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• pp.4280-4295
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• 2022

In this paper, a new multi-group neutron-gamma transport calculation code system STRAUM-MATXST for complicated geometrical problems is introduced and its development status including numerical tests is presented. In this code system, the MATXST (MATXS-based Cross Section Processor for S_N Transport) code generates multi-group neutron and gamma cross sections by processing MATXS format libraries generated using NJOY and the STRAUM (S_N Transport for Radiation Analysis with Unstructured Meshes) code performs multi-group neutron-gamma coupled transport calculation using tetrahedral meshes. In particular, this work presents the recent implementation and its test results of the Krylov subspace methods (i.e., Bi-CGSTAB and GMRES(m)) with preconditioners using DSA (Diffusion Synthetic Acceleration) and TSA (Transport Synthetic Acceleration). In addition, the Krylov subspace methods for accelerating the energy-group coupling iteration through thermal up-scatterings are implemented with new multi-group block DSA and TSA preconditioners in STRAUM.

• Nuclear Engineering and Technology
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• v.52 no.3
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• pp.485-498
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• 2020

A diffusion synthetic acceleration (DSA) technique for the S_N transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.

• International Journal of Air-Conditioning and Refrigeration
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• v.10 no.3
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• pp.162-175
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• 2002

To investigate smoke movement with radiation in a room fires, a numerical and experi-mental analysis were performed. In this paper, results from a field model based on a self-developed SMEP (Smoke Movement Estimating Program) were compared with Stockier's ex-periment and the experiments on various sized pool fires in a room with door The SMEP using PISO algorithm solves conservation equations for mass, momentum, energy and species, together with those for the modified k- $\varepsilon$ turbulence model with buoyancy term. Also it solves the radiation equation using the S-N discrete ordinates method (DOM). The result of the cal-culated smoke temperature considering radiation effect has shown good agreement compared with the experimental data, although there are large discrepancy in the hot smoke layer be-tween the temperature predicted by the SMEP with only convection effect and obtained by the experimental result. This large discrepancy is caused from the radiation effect of $H_2O$ and $CO_2$ gas under smoke productions. Hence the radiation effect under smoke in fire is the point to be specially considered in order to produce more realistic result.