随机中子输运方程数值解与近似解的对比分析

在描述具有弱源的有限增殖系统引发持续裂变链概率时,随机中子输运理论点堆模型近似解和考虑中子位置和速度方向(r,v)相空间模型的数值解的形式不同,用一个与球形堆半径r相关的改进点堆模型近似解与Sn方法数值解进行比较,由此可以看出两者的差异及非线性项的影响程度.     

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.     

Progress in nodal methods for the solution of the neutron diffusion and transport equations

Recent progress in the development of coarse-mesh nodal methods for the numerical solution of the neutron diffusion and transport equations is reviewed. In contrast with earlier nodal simulators, more recent nodal diffusion methods are characterized by the systematic derivation of spatial coupling relationships that are entirely consistent with the multigroup diffusion equation. These relationships most often are derived by developing approximations to the one-dimensional equations obtained by integrating the multidimensional diffusion equation over directions transverse to each coordinate axis. Both polynomial and analytic approaches to the solution of the transverse-integrated equations are discussed, and the Cartesian-geometry polynomial approach is derived in a manner which motivates the extension of this formulation to the solution of the diffusion equation in hexagonal geometry. Iterative procedures developed for the solution of the nodal equations are discussed briefly, and numerical comparisons for representative three-dimensional benchmark problems are given.

The application of similar ideas to the neutron transport equation has led to the development of coarse-mesh transport schemes that combine nodal spatial approximations with angular representations based on either the standard discrete-ordinate approximation or double P_n expansions of the angular dependence of the fluxes on the surfaces of the nodes. The former methods yield improved difference approximations to the multidimensional discrete-ordinates equations, while the latter approach leads to equations similar to those obtained in interface-current nodal-diffusion formulations. The relative efficiencies of these two approaches are discussed, and directions for future work are indicated.     

中子积分输运动态方程

通过理论分析给出了中子积分输运动态方程 ,发展了中子积分输运理论 ,使中子积分输运理论不仅可以用来分析反应堆栅格非均匀效应和计算反应堆参数等稳态问题 ,而且还可以处理反应堆动态问题。中子积分输运动态方程是一个多群多点 (一个空间分区为一点 )中子动态方程 ,在单群情况下就是多点反应堆动态方程。多点动态方程可以用来分析与空间有关的反应堆动态问题。介绍了中子积分输运动态方程的应用个例 ,通过中子积分输运动态方程分析了中国先进研究堆中子代时间的构成 (刚性和柔性中子代时间 )问题。     

An integral equation arising in neutron transport theory

A classic problem in nuclear reactor physics is the calculation of the spatial distribution of fissile material to make the associated neutron flux distribution spatially constant. We examine a special case of that problem for an infinite slab of fissile material which is infinitely reflected on both sides by a non-multiplying material. The conditions for a constant flux are derived and lead to a singular integral equation. This equation is reduced analytically to a non-singular integral equation and the solution thereby obtained is compared with that from a direct numerical method. Some of the physical implications are examined. We also note that, contrary to a theorem for multi-group diffusion theory, the resulting total fissile loading of the system is not a minimum but rather a maximum. An important aspect of the present work is that transport theory is used and not diffusion theory. Indeed, we note that no solution exists for the corresponding diffusion theory model unless it is specially modified by the addition of generalised functions, and hence we note that the problem is intrinsically governed by transport effects.     

On equiconvergence of ultraspherical polynomials solution of one-speed neutron transport equation

Ultraspherical polynomial approximation is used in slab criticality calculations for strongly anisotropic scattering. A unique and general formulation is developed for slab criticality condition whose sub-cases are spherical harmonics approximation, Chebyshev polynomial approximation of first and second kind. Since Legendre polynomials, Chebyshev Polynomials of first and second kinds are special cases of ultraspherical polynomials; our formulation inherently covers all these approximations and lets one to employ any other ultraspherical polynomial approximation in the solution of one-speed neutron transport equation. Our calculations showed that solution of one-speed neutron transport equation for various degrees of anisotropy and cross-section parameters is almost insensitive to the choice of ultraspherical polynomial with the present days’ computing capabilities. In other words, as much as high order ultraspherical polynomial approximation is used the solution converges to the same value for a specified problem regardless the type of the ultraspherical polynomial assigned in the solution as equiconvergence theorem of Jacobi polynomials states.     

On the solution of the dispersion equation for monoenergetic neutron transport with linearly anisotropic scattering

The eigenvalues occurring in the stationary or time dependent, monoenergetic Boltzmann equation with linearly anisotropic scattering are investigated. A detailed analysis is made of the number, nature and behaviour of the eigenvalues for the stationary case. This is applied to an almost identical equation obtained by Williams in the theory of slowing down of particles. In the time dependent case, a semi-analytical proof is given for the existence of complex eigenvalues, which are not encountered for isotropic scattering.     

On the use of the conjugate gradient method for the solution of the neutron transport equation

The possibility of using the standard conjugate gradient (CG) method to directly solve the S_n-equations based on the diamond difference scheme is studied for mono-energetic neutron transport problems with isotropic scattering. It is shown that such a direct use is possible for practical heterogeneous problems with a significant speed-up over the conventional source iteration (SI) method except for the problems that are prone to unphysical negative fluxes. Some recipes are suggested to make use of the CG-method even in those cases which need negative flux fix-up in the SI-method. The transport synthetic acceleration scheme, recently developed by Ramone [Nucl. Sci. Eng. 125 (1997) 257] and others, is shown to be useful in such cases. A symmetrisation scheme for the coefficient matrix has also been presented to enable the use of the CG-method. This scheme is compared with another approach of using weighted inner products.     

A new integral method for solving the point reactor neutron kinetics equations

A numerical integral method that efficiently provides the solution of the point kinetics equations by using the better basis function (BBF) for the approximation of the neutron density in one time step integrations is described and investigated. The approach is based on an exact analytic integration of the neutron density equation, where the stiffness of the equations is overcome by the fully implicit formulation. The procedure is tested by using a variety of reactivity functions, including step reactivity insertion, ramp input and oscillatory reactivity changes. The solution of the better basis function method is compared to other analytical and numerical solutions of the point reactor kinetics equations. The results show that selecting a better basis function can improve the efficiency and accuracy of this integral method. The better basis function method can be used in real time forecasting for power reactors in order to prevent reactivity accidents.     

Numerical solution of neutron kinetics equations using A-stable algorithms

This paper gives a detailed account of both theoretical and numerical investigations which have been conducted in the application of A-stable algorithms to neutron kinetics problems. It is broadly divided into three sections. General considerations on desirable features of a reactor dynamics code are followed by the theoretical background. In order to be self-contained, the stability properties of one-step methods are recalled with emphasis on the A-stability concept introduced by Dahlquist. An algorithm is described, based on the interpolation of exp(z) in the unit disc of the complex plane, which generates A-stable schemes w_nn(z), (n= 1,…) with so-called ‘spectral matching’ properties. Practical reasons limit to w₁₁ (z) its use for the integration of the kinetics equations and the analytical properties of this first order rational approximation to the exponential function are studied. A second class of suitable integration schemes is made of the implicit Runge-Kutta (IRK) family, particularly the subclass of diagonally implicit Runge-Kutta (DIRK) methods which are factorizable. Finally, the numerical results obtained with these algorithms are discussed on a set of four point kinetics problems for both fast and thermal-type reactors.     

Numerical solution of the neutron transport equation using cellular neural networks

Various methods have been used for solving the neutron transport equation in the past, and a number of computer codes have been developed based on these solution methods. This paper describes a novel method for the solution of the steady-state and time-dependent neutron transport equation using the duality between neutronic parameters in the method of characteristic (MOC) and the electrical parameters in the cellular neural networks (CNN). The relevant electrical circuit can be simulated by professional electrical circuit simulator software, HSPICE. This software is used for numerical solution of the transport equation only by preparation of appropriate inputs. This method does not need inner and outer iterations, which is a necessary step in the other deterministic methods. One of the main applications of the proposed method may be the development of a new hardware by VLSI technology for online spatio-temporal calculations of the transport equation for nuclear reactor core. The accuracy and capability of this method are examined in a 2D steady-state problem for a BWR fuel assembly, and a 2D time-dependent TWIGL seed/blanket problem.     

Bilinear-discontinuous numerical solution of the time dependent transport equation in slab geometry

A bilinear-discontinuous (BLD) discretization in space and time is described for the numerical solution of the discrete ordinates form of the time dependent, one speed neutral particle transport equation in slab geometry. Numerical results using the BLD method are compared with analytical approximations and other space-time discretization schemes. Analysis shows the BLD method is stable and third order accurate in the space and time dimensions independently.     

Rigorous solution to multiregion problems of monoenergetic neutron transport theory

A rigorous semianalytical algorithm is used, in the frame of the integral form of the transport equation, for the solution of some basic multilayer problems of monoenergetic neutron transport theory. The critical problem for a three region reactor is explicitly worked out, and numerical results are presented, in comparison with F_N calculations.     

A perturbation theory for the non-linear system of neutron transport and burnup equations

A perturbation theory for use in nuclear reactor burnup analysis is derived. An important characteristic function is defined, and the related adjoint equation is obtained simply by using the variational principle. The adjoint matrix operator is evaluated directly from its defining differential expression. Responses at the end of cycle due to changes in initial material inventory, in nuclear data, and in power demands can be calculated using the previously determined forward and adjoint solutions. The method has additional applications, notably in selecting beginning of cycle conditions so as to achieve a particular end of cycle condition.     

An exact special case solution of the neutron stochastic equations at low power

An exact solution for arbitrary physical data of the neutron chain multiplication problem is given for the special case of prompt neutron stochastics in a steady-state sub-critical assembly. These exact solutions enable a comparison of several approximations to be made including a new version of the continuum distribution, correct to all moments. In general, the approximate models are correct to second order moments of the distribution of neutrons. It is shown that the effect of third and higher order moments is generally negligible but that the ‘extinction’ probability may be substantially in error in the various approximations.     

Milne’s half-space problem: A numerical solution of the related integral equation

The one speed, isotropic scattering, planar Milne’s problem in neutron transport has been a test bed for many methods. We further explore here a numerical solution of the related integral equation. We find that with the use of singularity subtraction, high order quadratures, some interval transformations, and very high precision in all computations, it is possible to obtain results of benchmark accuracy in a straightforward manner.