ARES is a multidimensional parallel discrete ordinates particle transport code with arbitrary order anisotropic scattering. It can be applied to a wide variety of radiation shielding calculations and reactor physics analysis. ARES uses stateoftheart solution methods to obtain accurate solutions to the linear Boltzmann transport equation. A multigroup discretization is applied in energy. The code allows multiple spatial discretization schemes and solution methodologies. ARES currently provides diamond difference with or without linearzero flux fixup, theta weighted, directional theta weighted, exponential directional weighted, and linear discontinuous finite element spatial differencing schemes. Discrete ordinates differencing in angle and spherical harmonics expansion of the scattering source are adopted. First collision source method is used to eliminate or mitigate the ray effects. Traditional source iteration and Krylov iterative method preconditioned with diffusion synthetic acceleration are applied to solve the linear system of equations. ARES uses the KochBakerAlcouffe parallel sweep algorithm to obtain high parallel efficiency. Verification and validation for the ARES transport code system have been done by lots of benchmarks. In this paper, ARES solutions to the HBR2 benchmark and C5G7 benchmarks are in excellent agreement with published results. Numerical results are presented which demonstrate the accuracy and efficiency of these methods.
Particle transport problems arise in many different areas of engineering physics. There are two main types of simulation approaches in particle transport modeling: stochastic (Monte Carlo) and deterministic [
ARES [
Particle transport is an extremely challenging computational problem since the governing equation is sixdimensional with a high degree of coupling between these variables. A multigroup discretization is used in energy. Discrete ordinates differencing in angle and spherical harmonics expansion of the scattering source are adopted. The angular variable is usually discretized by replacing angular integrals with quadrature sums. The code allows multiple spatial discretization schemes and solution methodologies. A variety of spatial differencing scheme options are available, including diamond difference with or without linearzero flux fixup, theta weighted, directional theta weighted, exponential directional weighted, and linear discontinuous finite element scheme. This discretization produces a large, sparse, linear system of equations in a sevendimensional phase space.
The most general solution technique is source iteration. However, for optically thick problems dominated by scattering, the source iteration method converges very slowly. ARES uses the KochBakerAlcouffe parallel sweep algorithm to obtain high parallel efficiency.
Since discrete ordinates method was put forward by Carlson, many computer codes based on
This paper is organized as follows. In Section
The Boltzmann transport equation solved in ARES is
As we can see, the state is determined by the angular flux
Each independent variable needs to be discretized in deterministic solutions. The most commonly used angular discretization is the discrete ordinates method. This discretization produces a large, sparse, and linear system of equations in a sixdimensional phase space. We now briefly discuss each of these discretizations.
In
Quadrature sets are comprised of discrete directions and associated weight coefficients. For 3D geometry, ARES provides the level symmetric quadrature sets, the equal weight quadrature sets, and the evenodd moment quadrature sets, which are fullsymmetric, as well as the LegendreChebyshev quadrature sets which are halfsymmetric. And a biasing technique based on the LegendreChebyshev quadrature sets, called “the angular refinement technique for polar angles,” is developed.
For 3D transport calculation, the level symmetric quadrature sets (LS or
Point arrangement for
From the point of view to satisfy more moment conditions for integration accuracy, the evenodd moment quadrature sets (
The halfsymmetric LegendreChebyshev quadrature sets (
Point arrangement for
For problems that angular flux distribution or spatial distribution of scalar flux is highly peaked caused by highly anisotropic scattering or regional materials, results of transport calculation with traditional quadrature sets may be unsatisfying. To solve this problem, we designed a quadrature’s biasing technique called “the angular refinement technique for polar angles” based on the
ARES currently provides diamond difference with or without linearzero flux fixup, theta weighted (TW), directional theta weighted (DTW), exponential directional weighted (EDW), and linear discontinuous finite element spatial differencing schemes.
The diamond difference method, which assumes a linear relationship between the directional flux at the cell center and cell boundaries, is simple and accurate for small mesh intervals. When the mesh interval is too large, the difference equations may yield negative fluxes. The TW, DTW, and EDW variations on the DD method were developed to eliminate the appearance of negative fluxes without significantly sacrificing computational cost or accuracy.
The balance equation can be obtained by integrating the discretized form of the transport equation over the mesh cell such as
The accuracy of the diamond difference scheme is secondorder truncation. Considering that negative boundary angular flux is nonphysical, the negative flux set to zero fixup is commonly used. However, the fixup causes DZ to become nonlinear and depart from secondorder accuracy [
The scheme uses the incoming fluxes to calculate weighting factors. The cellcentered and exiting fluxes vary smoothly between the step and diamond difference approximations. The weighting factors are calculated from the following system of equations:
The thetaweighting factors
The directional theta weighted scheme is an extension of TW scheme. The directionbased parameters are used to obtain angular flux weighting factor. To be consistent, the weights (
The EDW scheme uses the DTW to predict a solution that is then corrected by an exponential fit and it should be more stable and accurate than DTW alone. We simply write down the equations that are solved in ARES. The inherently positive exponential auxiliary equations are given. The exponential coefficients
This method is absolutely positive, stable, and directionally weighted and is significantly more accurate than the DTW scheme in streaming problems with relaxed cell intervals [
Discontinuous finite element differencing captures discontinuities in solution and material properties and has thirdorder accuracy for global quantities, is acceleratable and damped, and has the diffusion limit [
ARES uses traditional source iteration and Krylov methods to solve transport equation. To make the discussion of numerical solution techniques clear, discretized transport equation can be expressed in operator notation [
The standard way to calculate eigenvalue is power iteration [
Within each power iteration, the method for solving (
More specifically, with the groups defined over the energy range
When using GaussSeidel iteration over energy, one must solve
ARES provides source iteration and Krylov for the withingroup equations. Source iteration can be thought of as a twopart process:
Here,
However, as the problem becomes more scatteringdominated, source iteration will be increasingly inefficient. Classic diffusion synthetic acceleration scheme suffers from severe stability problems in three dimensions with large material discontinuities [
The desired form for Krylov iteration is given as follows:
Krylov iteration schemes are particularly amenable for this quite large, fairly sparse matrix because only the action of operator
The problems typically of interest in the nuclear engineering community are of large scale. As larger computer resources have made it possible, some discrete ordinates codes on massive parallel machines, such as PARTISN and Denovo, have been developed. Motivated by the required ability to calculate large scale problems on available computer resources, ARES is developed with capability of paralleling.
The efficiency of the discrete ordinates method is largely dependent on the efficiency of the transport sweep procedure. KBA algorithm [
Decomposition of 3D structured mesh for KBA algorithm.
For a given direction, the KBA algorithm orders the tasks as depicted in Figure
KBA sweep ordering.
The discrete directions are pipelined so that the sweep along the next direction in the octant begins without waiting, when sweeping along the current direction is completed. The pipelining can be visualized as repeating the domains
We note that some processors remain idle at the beginning and ending of pipeline, which causes PCE is less than unity. Some more gains could be obtained by sweeping on all quadrants at once rather than sequentially [
Ray effects, shown as unphysical oscillations in the scalar flux, are inherent problem in
The first collision source method decomposes the transport equation into (
The uncollided flux moments are calculated analytically by (
In ARES, arbitrary number of point sources are employed to approximate the source region. And the location of a point source can be arbitrary. RAY employs ray tracing method to accelerate the calculation of optical distances
The ARES code system has been validated and verified by lots of analytical problems, international benchmark problems, and international benchmark experiments. All results of ARES have been compared with authoritative transport codes, such as TORT and MCNP.
The Kobayashi benchmark problem [
The Takeda benchmark problems [
In this chapter, the HBR2 benchmark problem [
An accurate calculation of the neutron fluence at the reactor vessel is necessary to estimate the structural integrity over the designed lifetime and to support analyses for a potential plant life extension. The H. B. Robinson Unit 2 Pressure Vessel Benchmark (HBR2 benchmark), based on experimental data from an operating PWR reactor, is the only openly available RPV benchmark through the SINBAD Database at the OECD/NEA Data Bank [
HBR2 benchmark is a 2300 MW (thermal power) pressurized water reactor designed by Westinghouse, as shown in Figure
Horizontal cross section of the HBR2 reactor. 1: reactor core; 2: core baffle; 3: core barrel; 4: downcomer reg. #1; 5: thermal shield; 6: downcomer reg. #2; 7: pressure vessel; ×: capsule dosimeters location.
To describe the model accurately, the calculation of neutron source is significant. The power to neutron source conversion factor was calculated based on the contributions of ^{235}U and ^{239}Pu to the fission neutron source, and we took the average fission spectrum of ^{235}U and ^{239}Pu as the source energy spectrum [
The MUSE1.0 [
With the purpose of comparing the calculated and measured specific activities, we take the reactor power changes during irradiation period and the affection of the closest fuel assemblies to the reaction rate at the dosimetry locations into account. According to the relative reactor data given by the benchmark, the approximate reaction rate can be written as
For all the considered typical nuclide reaction, the reaction rates of the dosimeters in the surveillance capsule calculated by ARES, the reference values provided by DORT for the cycle average power distribution, and core power of 2300 MW are given in Table
Reaction rates calculated by ARES and reference values (DORT).
Reaction rate (s^{−1}atom^{−1})  







Reference (DORT) 





ARES 





According to the reaction rates from Table
Calculated specific activities.
Specific activity (Bq/mg)  








30 years  71 days  313 days  84 days  5.3 years 






Reference (DORT) 





ARES 





The ratios of the calculated and measured specific activities are listed in Table
Ratios of calculatedtomeasured (





Average  

Reference 
0.85 
0.96  0.93  0.85  0.90 
0.90 ± 0.04 
ARES  0.82 
0.97  0.93  1.03  1.01 
0.95 ± 0.04 
The results indicate that the ARES transport calculation and the measured specific activities are in good agreement except
The C5G7 benchmark problem [
Geometry specification for C5G7 benchmark.
Mesh over a pin cell.
This problem is executed using an EO_{16} quadrature set with a total of 288 angles. Three cases characterized by control rods position are considered: (1) UNRODDED, control rods stay in the moderator above fuel assemblies, (2) RODDED A, the control rods are inserted onethird of the inner UO_{2} fuel assembly, and (3) RODDED B, the control rods are inserted twothird of the inner UO_{2} fuel assembly and onethird of the two MOX fuel assemblies. The convergence criteria were set to
Table
Comparison of calculated eigenvalue with reference value.
UNRODDED  RODDED A  RODDED B  

Reference MCNP 



ARES  1.14320  1.12826  1.07784 
Differences/pcm  10  18  6 
Figure
Relative differences of maximum pin power.
Assembly powers are calculated and compared with reference MCNP values in Table
Comparison of assembly power with reference value.
Assembly  Reference MCNP  ARES  Relative difference/% 

Unrodded inner UO_{2} 

491.26  0.012 
Unrodded MOX 

212.68 

Unrodded outer UO_{2} 

139.38 

Rodded A inner UO_{2} 

461.28 

Rodded A MOX 

221.69 

Rodded A outer UO_{2} 

151.34 

Rodded B inner UO_{2} 

395.59 

Rodded B MOX 

236.58 

Rodded B outer UO_{2} 

187.26 

In summary, the results show that ARES can be used to perform precise transport calculation for complex threedimensional geometries that include strongly absorbing materials, such as fuel assemblies without homogenization.
ARES code system is developed to solve the linearized Boltzmann transport equation for a wide variety of radiation transport applications and reactor physics analysis. ARES provides five spatial differencing schemes, uses the first collision source method to eliminate or mitigate the ray effects, and applies source iteration and Krylov iterative methods to solve the linear system of equations.
In this paper, ARES solutions to the HBR2 benchmark and C5G7 benchmarks are in excellent agreement with reference results. ARES is undergoing continuous development with many new features planned for implementation, aiming to deal with advanced physical model and problems of extended range, such as more accurate solutions for deep penetration problems with void region.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by National Natural Science Foundation of China (11505059 and 11575061) and the Fundamental Research Funds for the Central Universities (2016MS59).