Source Iteration Research Articles

ACS Algorithm in Discrete Ordinates for Pressure Vessel Dosimetry

The Adaptive Collision Source (ACS) method can solve the Linear Boltzmann Equation (LBE) more efficiently by adaptation of the angular quadrature order. This is similar to, and essentially an extension of, the first collision source method. Previously, the ACS methodology has been implemented into the TITAN discrete ordinates code, and has shown speedups of 2-4 on a simple test problem, with very little loss of accuracy (within a provided adaptive tolerance). This work examines the use of the ACS method for a more realistic problem: pressure vessel dosimetry with the VENUS-2 MOX-fuelled reactor dosimetry benchmark. The ACS method proved to be able to obtain accurate results while being approximately twice as efficient as using a constant quadrature in a standard source iteration scheme.

Corner neutronic code

The neutronic code CORNER is based on the SN discrete ordinates method [1] and the PM scattering cross-section approximation. It is intended for high-precision deterministic neutronic calculations of fast-neutron reactors and can be used to solve two types of steady-state neutron transport and gamma quanta problems in a 3D hexagonal geometry: Keff problems (homogeneous) and source problems (inhomogeneous).The code is developed in the Fortran language and has a modular structure. Its key modules are a module for the preparation of neutron constants in the ANISN format; a geometric module containing a description of the core's loading map and fuel assembly types, including their axial meshing and material composition; a module for preparing angular quadrature sets; an input data module containing the parameters of the approximation used and the control parameters; a neutronic calculation module and a calculation data processing module.The Directional Theta-Weighted (DTW) difference scheme [2] has been built to approximate the spatial dependence. It has advantages over the DD (Diamond Difference) scheme broadly used in coarse-mesh problems.The energy dependence is represented by multigroup approximation. The angular variable is discretized by introduction of the angular quadrature set. Quadrature sets can be also defined by the user.An iterative solution process is used, including external iterations for the fission source and internal iterations for the scattering source. The paper presents the results of a cross-verification against the Monte Carlo MMK code [3] and on a model of the BN-800 reactor core.

A correction and a clarification to “Optimizing variance reduction in Monte Carlo eigenvalue calculations that employ the source iteration approach”

Recently two papers were published proposing a way of efficiently estimating local responses within Monte Carlo eigenvalue calculations that employ the source-iteration scheme. This note corrects and clarifies some of the nomenclature in that work.

Three-dimensional simulations of MHD generator coupling with outer resistance circuit

The authors conduct fully-developed three-dimensional numerical simulations of a MHD generator coupling with outer resistance circuit. Instead of considering the MHD generator as a constant voltage source or a constant flow source, a simple and effective iteration is provided and successfully practiced to simulate the interaction between the MHD generator and the outer circuit so that they obey the Ohm’s Law and the Kirchhoff’s Law with sufficient accuracy set by user. The objective of this research is to have more detailed knowledge of the characteristics of MHD generators with incompressible medium, including the spatial variances of velocity and electromagnetic variables. In the calculations, two significant non-dimensional parameters Hac and Rec, whose influences on velocity and electromagnetic variables are elaborately discussed, are deduced from the governing equations. In general, Hac, proportional to the applied magnetic field, has negative effects on the variables discussed but Rec, corresponding to the driven pressure, has opposite effects.

Modified uniform-fission-site algorithm in Monte Carlo simulation of reactor criticality problem

Because of a very non-uniform power distribution in core region, a very non-uniform distribution of relative uncertainties exists for tallies in Monte Carlo criticality calculations of pin-by-pin reactor model. To make a large part of cells obtain small enough relative uncertainties with reasonable time costs, increasing the total sample scale is not a good choice. By realizing a modified uniform-fission-site algorithm on the basis of source iteration algorithm of parallel Monte Carlo transport code JMCT, we obtain higher efficiencies for tallies in the calculations of pin-by-pin model of the Dayawang reactor plant. This work supplies a useful tool for matching the goal of simulating the benchmark pin-by-pin reactor models with a pre-described standard(the so called Kord-Smith challenge).

Accelerative iteration for coupled conductive–radiative heat transfer computation in semitransparent media

Due to the intensive coupling effect and non-linear characteristic, coupled conductive–radiative heat transfer problem in semitransparent media always consumes an enormous amount of computing time. In present work, an accelerative iteration model is developed to speed up the convergence process of this kind of coupled calculation. It promotes the iterative efficiency by revising the temperature and incident radiation of media during the iteration of energy equation and radiative transfer equation. It can remarkably reduce the computing time of coupled conductive–radiative heat transfer calculation in semitransparent media, even in the cases with complicate boundary conditions. The accuracy of present model is verified by comparing with literature and traditional source iteration method. The influences of conduction–radiation parameter, optical thickness, scattering albedo, emissivity of walls and boundary conditions are investigated as well. The results indicate that the accuracy of present model is reliable and its accelerative effect is more significant for the optically thick and scattering-dominated media with complicate boundary conditions.

CMFD and GPU acceleration on method of characteristics for hexagonal cores

Coarse Mesh Finite Difference (CMFD) has been widely adopted as an effective way to accelerate the source iteration of transport calculation. However in a core with hexagonal assemblies there are non-hexagonal meshes around the edges of assemblies, causing a problem for CMFD if the CMFD equations are still to be solved via tri-diagonal matrix inversion by simply scanning the whole core meshes in different directions. To solve this problem, we propose an unequal mesh CMFD formulation that combines the non-hexagonal cells on the boundary of neighboring assemblies into non-regular hexagonal cells. We also investigated the alternative hardware acceleration of using graphics processing units (GPU) with graphics card in a personal computer. The tool CUDA is employed, which is a parallel computing platform and programming model invented by the company NVIDIA for harnessing the power of GPU. To investigate and implement these two acceleration methods, a 2-D hexagonal core transport code using the method of characteristics (MOC) is developed. A hexagonal mini-core benchmark problem is established to confirm the accuracy of the MOC code and to assess the effectiveness of CMFD and GPU parallel acceleration. For this benchmark problem, the CMFD acceleration increases the speed 16 times while the GPU acceleration speeds it up 25 times. When used simultaneously, they provide a speed gain of 292 times.

A Least-Squares Transport Equation Compatible with Voids

Standard second-order self-adjoint forms of the transport equation, such as the even-parity, odd-parity, and self-adjoint angular flux equation, cannot be used in voids. Perhaps more important, they experience numerical convergence difficulties in near-voids. Here we present a new form of a second-order self-adjoint transport equation that has an advantage relative to standard forms in that it can be used in voids or near-voids. Our equation is closely related to the standard least-squares form of the transport equation with both equations being applicable in a void and having a nonconservative analytic form. However, unlike the standard least-squares form of the transport equation, our least-squares equation is compatible with source iteration. It has been found that the standard least-squares form of the transport equation with a linear-continuous finite-element spatial discretization has difficulty in the thick diffusion limit. Here we extensively test the 1D slab-geometry version of our scheme with respect to void solutions, spatial convergence rate, and the intermediate and thick diffusion limits. We also define an effective diffusion synthetic acceleration scheme for our discretization. Our conclusion is that our least-squares Sn formulation represents an excellent alternative to existing second-order Sn transport formulations.

On the Consistent Spatial Homogenization Method in Neutron Transport Theory

The recently developed consistent spatial homogenization method in neutron transport theory is based on basis functions in spatial and angular domains and spatial rehomogenization to correct for the core environment effects while reproducing the heterogeneous solution in full phase space resolution and accuracy. In this article, an efficient version of the method is developed that also simplifies implementation and memory requirements for data handling in multidimensional geometries. This is done by (1) replacing the Fourier series with B-spline basis functions for the spatial domain; (2) truncating the angular expansion to first order, and (3) developing a source iteration method to replace the original local fixed-source transport method for on-the-fly rehomogenization. The new method is tested in a 1D benchmark problem characteristic of pressurized water reactors with mixed oxide fuel. The method is found to be very accurate and more efficient than the original consistent spatial homogenization method.

Optimizing variance reduction in Monte Carlo eigenvalue calculations that employ the source iteration approach

The question of variance reduction within the source-iteration scheme of a Monte Carlo eigenvalue calculation is tackled. The trade-off point between improving the statistics of a local response whilst simultaneously not damaging excessively the fundamental mode, the source for the calculation of the local response in the next fission cycle, is found. It is realized that applying less normalizations, i.e. employing superhistories, is advantageous. Realistic test problems, both fast and thermal fission, with in- and ex-core local responses, are treated. There is a good agreement between the predicted and actual gain in efficiency. A single comparison with the classic formalism, based purely on ϕ∗, shows large differences.

Moment-Based Acceleration for Neutral Gas Kinetics with BGK Collision Operator

In this work, we present a moment-based accelerator algorithm for a Picard iteration applied to a neutral gas dynamics Boltzmann transport equation with a Bhatnagar-Gross-Krook collision operator. Traditional approaches relying on either explicit or Picard iteration schemes (i.e., source iteration) are severely limited for investigating time-scales much larger than the collisional relaxation time-scale, τ. We have developed a nonlinear accelerator algorithm that allows one to step over this stiff collision time scale and follow the hydrodynamic time scale of the problem when appropriate. The new algorithm relies on formulating a nonlinear, coupled system comprised of a high-order (HO) kinetic equation and a low-order (LO) fluid moment equation system. The HO equation provides self-consistent closures to the LO fluid equations, while the latter provides the required implicit-moment variables to evaluate the collision operator. We characterize the performance of the new algorithm on a Sod shock tube and a strong shock tube problem with varying Knudsen number.

S2SA preconditioning for the Sn equations with strictly nonnegative spatial discretization

Preconditioners based upon transport sweeps and diffusion-synthetic acceleration have been constructed and applied to the zeroth and first spatial moments of the 1-D Sn transport equation using a strictly nonnegative nonlinear spatial closure. Linear and nonlinear preconditioners have been derived and analyzed. The effectiveness of various combinations of these preconditioners are compared using the source iteration, matrix-free Picard Krylov, and nonlinear Krylov acceleration methods. In one dimension, preconditioning with a linear S2SA diffusion equation is found to be essentially equivalent to using a nonlinear diffusion equation. The ability to use a linear diffusion equation has important implications for preconditioning the Sn equations with a strictly nonnegative spatial discretization in multiple dimensions.

Diffusion Acceleration Schemes for Self-Adjoint Angular Flux Formulation with a Void Treatment

A Galerkin weak form for the monoenergetic neutron transport equation with a continuous finite element method and discrete ordinate method is developed based on self-adjoint angular flux formulation. This weak form is modified for treating void regions. A consistent diffusion scheme is developed with P0 projection. Correction terms of the diffusion scheme are derived to reproduce the transport scalar flux. A source iteration that decouples the solution of all directions with both linear and nonlinear diffusion accelerations is developed and demonstrated. One-dimensional Fourier analysis is conducted to demonstrate the stability of the linear and nonlinear diffusion accelerations. Numerical results of these schemes are presented.

Coarse mesh finite difference formulation for accelerated Monte Carlo eigenvalue calculation

An efficient Monte Carlo (MC) eigenvalue calculation method for source convergence acceleration and stabilization is developed by employing the Coarse Mesh Finite Difference (CMFD) formulation. The detailed methods for constructing the CMFD system using proper MC tallies are devised such that the coarse mesh homogenization parameters are dynamically produced. These involve the schemes for tally accumulation and periodic reset of the CMFD system. The method for feedback which is to adjust the MC fission source distribution (FSD) using the CMFD global solution is then introduced through a weight adjustment scheme. The CMFD accelerated MC (CMFD-MC) calculation is examined first for a simple one-dimensional multigroup problem to investigate the effectiveness of the accelerated fission source convergence process and also to analyze the sensitivity of the CMFD-MC solutions on the size of coarse meshes and on the number of CMFD energy groups. The performance of CMFD acceleration is then assessed for a set of two-dimensional and three-dimensional multigroup (3D) pressurized water reactor core problems. It is demonstrated that very rapid convergence of the MC FSD is possible with the CMFD formulation in that a sufficiently converged MC FSD can be obtained within 20 cycles even for large three-dimensional problems which would require more than 600 inactive cycles with the standard MC fission source iteration scheme. It is also shown that the optional application of the CMFD formulation in the active cycles can stabilize FSDs such that the real-to-apparent variance ratio of the local tallies can be reduced. However, due to the reduced importance of the variance bias in fine local tallies of 3D MC eigenvalue problems, the effectiveness of CMFD in tally stabilization turns out to be not so great.

Parallel sweep-based preconditioner for the solution of the linear Boltzmann transport equation

The Boltzmann transport equation is solved in the context of radiative heat transfer, for an isotropically scattering medium with reflecting boundaries. Under these circumstances, the different ordinates of the angular flux are mutually coupled. We explore here the use of a parallel sweep-based block diagonal preconditioner as a complement of the GMRES solver on the solution of the discretization matrix (which includes all the inter-ordinate couplings). The validity of this approach, when compared to the standard source iteration scheme, is successfully assessed for a significant range of the coupling parameters.

An Lp theory for stationary radiative transfer

We present a self-contained analysis of the stationary radiative transfer equation in weighted spaces. The use of weighted spaces allows us to derive uniform a-priori estimates for under minimal assumptions on the parameters. By constructing an explicit example, we show that our estimates are sharp and cannot be improved in general. Better estimates are, however, derived under additional assumptions on the parameters. We also present estimates for derivatives and traces of the solution and formulate a natural energy space, for which the data-to-solution map becomes an isomorphism. As a side result, we are able to prove uniform convergence of the source iteration for all without the assumption of positive absorption that is frequently used in the literature.

A Cellwise Block-Gauss-Seidel Iterative Method for Multigroup SN Transport on a Hybrid Parallel Computer Architecture

A Fourier analysis is conducted in two-dimensional (2-D) geometry for the discrete ordinates (SN) approximation of the neutron transport problem solved with Richardson iteration (source iteration) using the cellwise block-Jacobi (bJ) and block-Gauss-Seidel (bGS) algorithms. The results of the Fourier analysis show that convergence of bJ can degrade, leading to a spectral radius ρ equal to 1, in problems containing optically thin cells. For problems containing cells that are optically thick, instead, ρ tends to 0. Hence, in the optically-thick-cell regime, bJ is rapidly convergent even for scattering-dominated problems. Similar conclusions hold for bGS, except bGS approaches the asymptotic, thick-cell regime at convergence rates higher than bJ. Hence, we have implemented the bGS algorithm on the Roadrunner hybrid, parallel computer architecture. A compute node of this massively parallel machine comprises AMD Opteron cores that are linked to a Cell Broadband Engine (Cell/B.E.). LAPACK routines have been ported to the Cell/B.E. in order to make use of its parallel synergistic processing elements (SPEs). The bGS algorithm is based on the LU factorization and solution of a linear system that couples the fluxes for all SN angles and energy groups on a mesh cell. For every cell of a mesh that has been parallel decomposed on the higher-level Opteron processors, a linear system is transferred to the Cell/B.E. and the parallel LAPACK routines are used to compute a solution, which is then transferred back to the Opteron, where the rest of the SN transport computations take place. Compared to standard parallel machines, a one-hundred-fold speedup of the bGS was observed on Roadrunner. Numerical experiments with strong and weak parallel scaling demonstrate that the bGS method is viable and compares favorably to full parallel transport sweeps (FPS) on 2-D unstructured meshes when it is applied to optically thick, multimaterial problems. Specifically, the strong parallel efficiency of accelerated bGS on Roadrunner can achieve 73% at 512 processors, compared with 32 processors, while efficiency is 34% for the (Opteron-only) implementation of FPS. The weak parallel efficiency of bGS is 58% while it reaches 10% for FPS. As expected, however, bGS is not as efficient as FPS in optically thin problems.

Iterative performance of various formulations of the [formula omitted] equations

In this paper, the Standard, Composite, and Canonical forms of the Simplified PN (SPN) equations are reviewed and their corresponding iterative properties are compared. The Gauss–Seidel (FLIP), Explicit, and preconditioned Source Iteration iterative schemes have been analyzed for both isotropic and highly anisotropic (Fokker–Planck) scattering. The iterative performance of the various SPN forms is assessed using Fourier analysis, corroborated with numerical experiments.

Hybrid Deterministic/Monte Carlo Neutronics

In this paper we describe a hybrid deterministic/Monte Carlo algorithm for neutron transport simulation. The algorithm is based on nonlinear accelerators for source iteration, using Monte Carlo methods for the purely absorbing high-order problem and a Jacobian-free Newton--Krylov iteration for the low-order problem. We couple the Monte Carlo solution with the low-order problem using filtering to smooth the flux and current from the Monte Carlo solver and an analytic Jacobian-vector product to avoid numerical differentiation of the Monte Carlo results. We use a continuous energy deposition tally for the Monte Carlo simulation. We conclude the paper with numerical results which illustrate the effectiveness of the new algorithm.

Nonlinear Acceleration of Transport Criticality Problems

We present a nonlinear acceleration algorithm for a transport criticality problem. The algorithm combines the well-known nonlinear diffusion acceleration (NDA) algorithm with a recently developed, Newton-based nonlinear criticality acceleration (NCA) algorithm. The algorithm first employs NDA to reduce the system to scalar flux, then NCA is applied to the resulting drift-diffusion system. We apply a nonlinear elimination technique to eliminate the eigenvalue constraint equation from the Jacobian matrix. Numerical results show that the algorithm can reduce the CPU time by a factor of 30 to 400 compared to traditional power iterations (PIs) combined with standard source iterations and by a factor of 3 to 5 compared to application of NDA combined with inner PIs.