Multigroup Neutron Transport Research Articles

The finite-element method for multigroup neutron transport: Anisotropic scattering in 1-D slab geometry

Proof-tests on 1-D multigroup neutron transport problems are reported for strong anisotropic scattering. These tests have been undertaken as part of the validation of the 3-D multigroup finite-element transport code fel tran for ansisotropic scattering media. To illustrate the treatment of within-group and intergroup anisotropic scattering in the finite-element method the relevant theory is outlined. Ingroup scattering is checked using the backward-forward-isotropic (BFI) scattering law for source and eigenvalue problems. With this law anisotropic scattering problems can be transformed into equivalent isotropic scattering problems. In this way the well-validated isotropic scattering version of fel tran is used to validate the anisotropic version. Intergroup scattering effects are checked by solving few-group source problems for P 1 and P 3 scattering and the BFI scattering law. For P 1 and P 3 scattering checks are made with the discrete-ordinate finite-difference code anisn and the spherical harmonics finite-difference code marc/pn. For the BFI scattering law comparison is made with two-group exact solutions of Williams (1985) for 1-D systems.

Spectral analysis of the multigroup transport operator

Abstract A spectral analysis is given of the linear operator for multigroup neutron transport in a one-dimensional homogeneous plane slab. Azimuthal symmetry and isotropic scattering are assumed. The spectrum consists of a continuous and a discrete part; the multiplicity is nonuniform throughout the spectrum. Particular emphasis is placed on the construction and representation of a diagonalizing transformation.

Three-dimensional nodal diffusion and transport methods for the analysis of fast-reactor critical experiments

An interface-current nodal formulation is developed for the solution of the multigroup neutron diffusion and transport equations in three-dimensional Cartesian geometry. The nodal diffusion method uses polynomial approximations equivalent to those used in the nodal expansion method (NEM), while the nodal transport method is based on the approximation of exact one-dimensional integral equations. The transport scheme employs a double P-1 expansion of the surface angular fluxes in conjunction with either isotropic or angle-dependent approximations to the angular dependence of the transverse-leakage terms. The transport formulation does not involve an iteration on the scattering source, and identical iterative strategies are used to solve the interface-current equations obtained in both methods. Numerical results for a ZPPR fast-reactor critical experiment show that the nodal diffusion method produces spatially-converged solutions with significant reductions in computational cost relative to the finite-difference method. Use of the nodal transport scheme eliminates over 80% of the error due to the use of diffusion theory, with computational costs within a factor of two of nodal diffusion theory.

An arbitrary geometry finite element method for multigroup neutron transport with anisotropic scattering

A variational finite element-spherical harmonics method is presented for the solution of the even-parity multigroup equations with anisotropic scattering and sources. It is shown that by using a simple and natural formulation the numerical implementation of the method for any desired geometry is greatly eased and the anisotropy of scatter treated without any difficulty. Numerical examples demonstrate the ability of the resulting code to solve geometrically complex multigroup problems.

Simulation of Multigroup X-Ray, Alpha-Particle and Neutron Transport in Ion Beam Driven ICF Target

Simulation of Multigroup X-Ray, Alpha-Particle and Neutron Transport in Ion Beam Driven ICF Target

Applications of the FN method to transport calculations

Abstract A variation of the FN method for which convergence is known is applied to calculation of Chandrasekhar's H-function and to exit distribution problems in multi-group neutron transport.

The Solution of the Multigroup Neutron Transport Equation Using Spherical Harmonics

A solution of the multigroup neutron transport equation in one, two, or three space dimensions is presented. The flux ϕg(r, Ω) at point r in direction Ω̄ for energy group g takes the form of an expansion in unnormalized spherical harmonics. Thus,,where θ and ϕ are the axial and azimuthal angles of Ω, the associated Legendre polynomials, and N an arbitrary odd number. Using the various recurrence formulas for , a linked set of first-order differential equations in the moments results.Terms with odd 1 are eliminated yielding a second-order system to be solved by two methods. First, a finite difference formulation using an iterative procedure is given, and second, in XYZ and XY geometry, a finite element solution is presented. Results for a test problem using both methods are exhibited and compared.

A comparison of finite elements and discrete ordinate methods for one-dimensional multigroup problems

The finite element formulation for the multigroup neutron transport equation is utilized to solve both shielding and eigenvalue problems in one dimension. In particular the variational principles formulated by Ackroyd are used. Results obtained for a three-group constant source and a five-group eigenvalue problem show the accuracy and efficiency of the finite element method in comparison with the discrete ordinates finite difference method.

A comparison of iteration schemes for Chandrasekhar H-equations in multigroup neutron transport

An iteration scheme for the Chandrasekhar H-equations in multigroup neutron transport is shown to converge to the solution of physical interest. Moreover, the convergence is more rapid than that provided by direct iteration.

The spectrum of the multigroup neutron transport operator for bounded spatial domains

The spectrum of the multigroup neutron transport operator A is studied for bounded spatial regions D which consist of a finite number of material subregions. Our main results provide simple conditions on the material cross sections which guarantee that (1) A possesses eigenvalues in the finite plane; (2) A possesses a ’’leading’’ eigenvalue λ0 which is real, not less than the real part of any other eigenvalue, and to which there corresponds at least one nonnegative eigenfunction ψλ0; and (3) A possesses a ’’dominant’’ eigenvalue λ0 which is real, simple, greater than the real part of any other eigenvalue, and whose eigenfunction ψλ0 satisfies ψλ0⩾0 and ∫ψλ0d2Ω≳0. We give examples to illustrate the results and to show that a leading eigenvalue need not be simple, nor its eigenfunction(s) positive.

On incorporating linear constraints intoH-matrix calculations

A method of computingH matrices appropriate to (for example) studies of multigroup neutrontransport theory and rarefied-gas dynamics is proposed.

Solution of multigroup neutron transport equation with account for discrete roots in region of continuous spectrum and multiple roots

Solution of multigroup neutron transport equation with account for discrete roots in region of continuous spectrum and multiple roots

Solution by iteration of H-equations in multigroup neutron transport

The Chandrasekhar H-equations for matrix-valued functions are solved by an iterative method. Complex variables and positivity techniques are used to obtain convergence. This approach may be applied to subcritical neutron transport in a slab with isotropic scattering.

Application of Finite Element Method to Two-Dimensional Multi-Group Neutron Transport Equation in Cylindrical Geometry

The finite element method is applied to the spatial variables of multi-group neutron transport equation in the two-dimensional cylindrical (r, z) geometry. The equation is discretized using regular rectangular subregions in the (r, z) plane. The discontinuous method with bilinear or biquadratic Lagrange's interpolating polynomials as basis functions is incorporated into a computer code FEMRZ. Here, the angular fluxes are allowed to be discontinuous across the subregion boundaries. Some numerical calculations have been performed and the results indicated that, in the case of biquadratic approximation, the solutions are sufficiently accurate and numerically stable even for coarse meshes. The results are also compared with those obtained by a diamond difference S n code TWOTRAN-II. The merits of the discontinuous method are demonstrated through the numerical studies.

Biased Selection of Particle Flight Directions as a Variance Reduction Technique in Monte Carlo Calculations

As a practical variance reduction technique applicable to Monte Carlo shielding calculations, the present article shows a new simple biased sampling technique on particle flight directions. Scattered particles not directed towards the detector positions are killed if they are not so important, that is, if the particle weights are sufficiently small compared to the source weight. In this way, we can reduce the sample size required for obtaining an accurate estimate for the detector response. The present technique was incorporated into the multigroup neutron and γ-ray transport code MORSE, and sample calculations were performed on spherical fast neutron systems. The results have shown that this biased technique is effective for dealing with neutron multiplication as well as neutron transmission problems. The neutron flux or the effective multiplication factor of a nuclear reactor is estimated more accurately than from the method of path-length stretching with about the same computation time. In addition, it...

The solution of the time-independent multi-group neutron transport equation using spherical harmonics

The time-independent multi-group neutron transport equation is solved by expressing the neutron flux for each energy group in the form Φ(R,Ω)= ∑ l=0 N ∑ m=0 l (2l+1)P m l( cosθ)(ψ lmr) cos(mφ)+γ lm(r sin(mθ)) where ф( r, Ω) is the flux at r in direction Ω. The moments ψ lm ( r) and y lm ( r) are functions of the position r and P m l (cos θ) is the associated Legendre polynomial of order lm. θ and ∅ are the spherical co-ordinates of Ω i.e. θ is the angle to the z-axis and ∅ the angle the projection of Ω on to the x− y plane makes with the x-axis. N is the order of the approximation. The resulting moment equations obtained after equating coefficients of P m l (cos θ) cos( m∅) and P m l (cos θ) sin ( m∅) are transformed, by eliminating ψ lm ( r) and y lm ( r) with odd l, to linked second order differential equations for those with even l. These are solved by the familiar algorithms of diffusion theory.

Application of Finite Element Method to Two-Dimensional Multi-Group Neutron Transport Equation in Cylindrical Geometry

The finite element method is applied to the spatial variables of multi-group neutron transport equation in the two-dimensional cylindrical (r, z) geometry. The equation is discretized using regular rectangular subregions in the (r, z) plane. The discontinuous method with bilinear or biquadratic Lagrange's interpolating polynomials as basis functions is incorporated into a computer code FEMRZ. Here, the angular fluxes are allowed to be discontinuous across the subregion boundaries. Some numerical calculations have been performed and the results indicated that, in the case of biquadratic approximation, the solutions are sufficiently accurate and numerically stable even for coarse meshes. The results are also compared with those obtained by a diamond difference S n code TWOTRAN-II. The merits of the discontinuous method are demonstrated through the numerical studies.

Steady, one-dimensional multigroup neutron transport with anisotropic scattering

The solution of steady, one-dimensional half-space multigroup transport problems with degenerate anisotropic scattering is obtained for L1 sources and incident distributions. The solution is expressed in terms of contour integrals of the resolvent operator (λI−K)−1, where K is the ’’separated’’ transport operator. The connection between this method and the ’’Case eigenfunction’’ method is briefly discussed, and the half-space albedo problem is treated in detail. This problem reduces to obtaining the Wiener–Hopf factorization of the dispersion matrix, hence to solving two coupled nonlinear, nonsingular matrix integral equations.

Multigroup neutron transport with degenerate kernel

Abstract The functional analytic method of Larsen and Habetler [1] is applied to the problem of multigroup neutron transport with degenerate kernel to obtain the eigensolutions of the reduced transport operator. This particular method yields a compact and mathematically rigorous solution of the problem. Applications to the full and half-range problems are indicated.

On k∞and other criticality conditions for an infinite system

Abstract It is shown that a study of certain class of matrices lead to criticality conditions based on the multigroup neutron transport model for a homogeneous infinite system. A consistent definition of k∞ is formulated. A simple example is presented.