Discrete Ordinates Neutron Transport Equation Research Articles

TARS: A parallel tetrahedral discontinuous finite element code for the solution of the discrete ordinates neutron transport equation

In this study, we introduce TARS, a neutron transport code based on the discrete-ordinates discontinuous finite element (SN-DFEM) method, which tackles existing challenges in the field by providing accurate and efficient solutions to the neutron transport equation. The use of DFEM enables higher-order spatial discretization, improving numerical stability, while parallelizing transport sweeping reduces computational time and resource requirements. TARS' performance was evaluated using the Takeda benchmark problems. The results showed that TARS generates highly accurate solutions, with eigenvalue errors no more than 90 pcm compared to MC solutions. The relative differences of region-averaged fluxes were below 1% in most regions, except those with severe flux gradients. Parallel calculation efficiency analysis revealed a PCE of over 70% even at 1024 CPU cores. However, hybrid MPI and OpenMP parallel programming did not outperform pure MPI programming, suggesting that the latter might be more suitable for TARS.

An analytical nodal method for solving the SN transport equation in 2-D Cartesian geometry

A novel Analytical Nodal Discrete Ordinates (ANDO) method for the solution of the discrete ordinates (SN) neutron transport equation in Cartesian geometry with isotropic scattering and constant neutron source is presented. A nodal method approximates the multi-dimensional transport equation as a system of coupled one-dimensional transport equations along each coordinate axis through transverse integration. The resulting transverse-integrated equations are then solved separately using a one-dimensional solution technique. The one-dimensional solution utilized in the new ANDO method is based on a recent closed-form analytical solution in slab geometry (WANG and BYAMBAAKHUU, 2017). The novel solution approach allows for a compact closed-form solution on each computational cell. Results for constant and linear transverse leakage approximations are presented. The ANDO method possesses a number of favorable properties, demonstrated with numerical results, such as high accuracy, rapid convergence, asymptotic preserving, positivity preserving, near linear computational complexity, and local-hp adaptivity. The ANDO method can also be easily extended to higher-order transverse leakage approximations, to three-dimensional Cartesian geometry, and to multi-group.

Mesh-free RBF collocation method to solve the multi-group SN neutron transport equation in arbitrary domains

Generally for numerical solution of the differential equations the mesh generation process is a major problem in classical mesh-based methods, especially in irregular domains. Contrary to the meshed based methods, Mesh Free (MFree) methods have been developed to avoid mesh generation. In the MFree methods a set of scattered nodes is used without requiring any mesh grid. So in this work the application of MFree collocation method is examined to solve SN neutron transport equation. The MFree collocation method uses the formulation of strong form which is truly mesh less method. Since MFree method is flexible for distributing arbitrary and irregular nodes, this method can be easily applied for high heterogeneous nuclear reactor problems with complicated geometry. To validate the proposed method some test problems including regular and irregular geometries have been modeled. Our results show that the mesh free collocation method can be used for discrete ordinate neutron transport equation as an efficient and accurate method without hardness and limitations of the mesh-based methods like finite element method.

Mesh-free method for numerical solution of the multi-group discrete ordinate neutron transport equation

The finite element method (FEM) is a widely-used method to solve neutron transport equation in an arbitrary domain, but in order to ensure the accuracy of solution a re-meshing process is often required and some regions of the domain need to be meshed with finer meshes and consequently the whole domain should be re-meshed again. To overcome this problem of the FEM we have applied Mesh Free (MFree) method for the solution of the neutron transport equation. In mesh free technique, there is no need to regenerate finer meshes to ensure the accuracy of the solution and only the number of nodes in local domains is increased. MFree method is flexible for distributing arbitrary and irregular nodes, therefore this method can be easily applied for high absorbent regions in the nuclear reactor problems. We have developed a program, named SNMF, which solves discrete ordinate neutron transport equation, using mesh free techniques based on Radial Point Interpolation Method (RPIM). To validate SNMF program some test problems have been modeled. For the first time, the application of MFree method is examined to solve neutron transport equation. Our results show that mesh free discrete ordinate method can be used as an efficient and accurate method to solve neutron transport equation without hardness and limitations of finite element methods.

Convergence Analysis of Additive Angular Dependent Rebalance Acceleration for the Discrete Ordinates Transport Calculations

In solving the discrete ordinates neutron transport equation, the additive angular dependent rebalance (AADR) acceleration method proposed by the authors previously is simple to implement, unconditionally stable, and very effective. For slab geometry problems, it is demonstrated via Fourier analysis that the spectral radii of the AADR acceleration in S4-like and DP1-like rebalances as well as DP0-like rebalance are less than that of diffusion synthetic acceleration (DSA). This AADR acceleration method is easily extendable to DPN-like and low-order SN-like rebalancing, and it does not require consistent discretizations between the high-order and low-order equations as does DSA. The continuous Fourier analysis is also performed for rectangular geometry. This Fourier analysis shows that the AADR with directional S2-like weighting functions, which uses two different rebalance factors for the x and y directions per octant, provides better results than the AADR with the normal S2-like weighting functions, which uses a single weighting function per octant. The low-order equation in AADR is solved by a preconditioned Bi-CGSTAB algorithm, which reduces computational burden significantly.

Application of preconditioned GMRES to the numerical solution of the neutron transport equation

The generalized minimal residual (GMRES) method with right preconditioning is examined as an alternative to both standard and accelerated transport sweeps for the iterative solution of the diamond differenced discrete ordinates neutron transport equation. Incomplete factorization (ILU) type preconditioners are used to determine their effectiveness in accelerating GMRES for this application. ILU( τ), which requires the specification of a dropping criteria τ, proves to be a good choice for the types of problems examined in this paper. The combination of ILU( τ) and GMRES is compared with both DSA and unaccelerated transport sweeps for several model problems. It is found that the computational workload of the ILU( τ)-GMRES combination scales nonlinearly with the number of energy groups and quadrature order, making this technique most effective for problems with a small number of groups and discrete ordinates. However, the cost of preconditioner construction can be amortized over several calculations with different source and/or boundary values. Preconditioners built upon standard transport sweep algorithms are also evaluated as to their effectiveness in accelerating the convergence of GMRES. These preconditioners show better scaling with such problem parameters as the scattering ratio, the number of discrete ordinates, and the number of spatial meshes. These sweeps based preconditioners can also be cast in a matrix free form that greatly reduces storage requirements.

Numerical analysis of step characteristic methods and computational comparison with step nodal methods in x, y geometry

Abstract We are pursuing a study of new numerical methods used to solve the x, y geometry discrete ordinates neutron transport equation. For both Ckl characteristic and Nkl nodal methods, the angular flux is approximated by a polynomial of total degree k inside each mesh cell and by polynomials of degree l along the cell edges. In this paper the convergence properties of the simplest methods of both families, i.e. the Ckl step characteristic and Nki step nodal schemes are investigated. On the one hand error bounds for a pure absorber calculation by the Ck0 characteristic methods are given. On the other hand experimentally observed errors and convergence rates are reported for the C00, C10, N00 and N10 schemes. It is proved that all the Ck0 methods are first order convergent in a discrete L2 norm. The numerical results confirm these theoretical estimations. In addition they indicate that the step nodal schemes are asymptotically second order convergent for unusually smooth data and solution, and asymptotic...

Error bounds for some characteristic methods in an exceptional case

Abstract New characteristic methods for the solution of the x,y geometry discrete ordinates neutron transport equation have recently been introduced. Five polynomials, without any continuity requirement are defined on each mesh cell. A polynomial of order k is used to approximate the angular flux inside the cell, while polynomial approximations of order l are used along the cell edges. Error bounds for a pure absorber calculation by this Ckl characsteristic method are given here for l lower or equal to 1 and for the simplest case. In this case, that we shall call the exceptional case, a uniform spatial mesh grid with rectangles of length △x and height △y is used, and the angular quadrature directions ω = (μ, ν) verify the condition (the characteristic lines are the diagonals of the cells). It is proved that, in a discrete L2 norm, thre Ckl method has a convergene rate equal to Min (k+2, 2l+1) for regular data and solution, and equal to Min for more realistic situations. We also provide some numerical resu...

Discrete Ordinate Neutron Transport Equation for Two-Dimensional Triangular Geometry by Finite Fourier Transformation

Discrete ordinate neutron transport equation is derived for two-dimensional triangular regions by applying the finite fourier transformation. Assuming that the angular flux on each side of a triangular region is constant and the source term is constant in a triangular region, a zero-th order discrete ordinate equation is derived. It is shown that in this equation, the angular flux can be obtained always positively, as long as the source is positive.

Discrete ordinate neutron transport equation for two-dimensional triangular geometry by finite fourier transformation.

Discrete ordinate neutron transport equation is derived for two-dimensional triangular regions by applying the finite fourier transformation. Assuming that the angular flux on each side of a triangular region is constant and the source term is constant in a triangular region, a zero-th order discrete ordinate equation is derived. It is shown that in this equation, the angular flux can be obtained always positively, as long as the source is positive.

On the Spatial Differencing of the Discrete Ordinate Neutron Transport Equation

A family of discontinuous weighted-residual approximation methods are formulated for the spatial differencing of the discrete-ordinate neutron transport equation. Several known difference schemes are obtained as special cases of the approximation procedures considered. Estimates pertaining to the stability and convergence of a subclass of the methods are derived.