Adjoint Transport Problem Research Articles

The adjoint transport problem applied to estimate neutral particle leakage in the discrete ordinates formulations

In source–detector problems, neutron leakage is a quantity of interest that could lead to improve shielding structures, thus reducing the dose received by humans. In this work, we apply an adjoint technique with spectral nodal methods to compute neutron leakage in multigroup one– and two–dimensional problems in the discrete ordinates (SN) formulation. The use of the adjoint technique to calculate the leakages due to various source distributions is very convenient as it is possible to run adjoint problems and store the neutron importance maps. Here we solve the homogeneous adjoint SN transport equation by considering unit outgoing adjoint flux at the boundary. In order to numerically solve slab– and X, Y–geometry problems, we use spectral nodal methods that have been widely applied and discussed in the literature. Numerical results are given to illustrate the present adjoint technique to estimate the neutron leakage for each energy group in source–detector problems. For all the test problems, the results obtained by the adjoint technique as described in this paper do agree with the results obtained by solving the analogous forward problem.

An Analytic Benchmark for Neutron Boltzmann Transport with Downscattering—Part II: Flux and Eigenvalue Sensitivities to Nuclear Cross Sections and Resonance Parameters

A benchmark to verify the accuracy of neutron transport criticality solvers along the energy dimension was established. For the first time, the analytic solution of the flux amplitude was derived in the particular case of an infinite-homogeneous medium with isotropic scattering in the center of mass and an arbitrary number of no-threshold, neutral particle reaction resonances (e.g., radiative capture, fission, and resonance scattering). In this paper, the benchmark is extended to the adjoint transport problem, and a solution to the adjoint flux is derived. The adjoint flux solution is then combined with the forward flux to obtain expressions for an arbitrary-order cross section and resonance parameter sensitivity coefficients. Finally, numerical solutions are provided for a benchmark problem constituted of the first resonance of 239Pu, the 6.67-eV resonance of 238U, and a scattering isotope with a flat cross section, allowing for computational verification of the sensitivity coefficients and nuclear data uncertainty of current neutron transport criticality codes. Through these novel results, this analytic benchmark can serve as a reference to verify the sensitivity analysis of neutron transport criticality calculations.

A spectral nodal method for the adjoint [formula omitted] neutral particle transport equations in [formula omitted]-geometry: Application to direct and inverse multigroup source-detector problems

In this work, we apply an adjoint technique to solve two-dimensional direct and inverse source-detector transport problems. In order to obtain the neutral-particle importance maps, we have developed the extended adjoint spectral Green’s function constant-nodal (SGF†-CN) method to numerically solve multigroup adjoint transport problems in the discrete ordinates formulation. Described here is the methodology to obtain the discretized SGF†-CN equations and the adjoint partial one-node block inversion scheme used to iteratively solve these equations. Then, the Levenberg–Marquardt method is applied to minimize a least-squares difference in order to identify the multigroup sources. To illustrate the accuracy of the present methodology, we present and discuss numerical results to two typical model problems.

The Adjoint Discrete Ordinates Solution Applied to Direct and Inverse Neutral Particle X, Y-Geometry Source-Detector Problems

Solving the adjoint transport equation allows obtaining maps of the neutral-particle importance to an objective function, e.g., a detector response. In source-detector problems, the adjoint interior source is defined as the macroscopic absorption cross section of the material the detector is made of. Determining the intensity of a source of particles in each energy group (causes) is possible by solving an inverse source-detector problem, given the source location and the detector response (effects). In this work, we present the application of an adjoint technique to solve energy multigroup X,Y-geometry direct and inverse transport problems. In order to obtain the importance maps, we have extended the adjoint spectral Green's function constant-nodal (SGF*-CN) method to numerically solve energy multigroup adjoint transport problems in the discrete ordinates formulation. Described here are the methodology to obtain the discretized SGF*-CNequations and the adjoint partial one-node block inversion scheme used to iteratively solve these equations. We present numerical results to two test problems to illustrate the accuracy of the present methodology.

Estimation of neutron sources driving prescribed power generations in subcritical systems using one-speed two-dimensional discrete ordinates formulations

It is well known that any subcritical system can be driven by time-independent interior sources of neutrons. Thus, we present a methodology to determine the intensities of uniform and isotropic sources of neutrons that must be added inside a subcritical system generating a prescribed steady-state distribution of power. To accomplish this, we use the time-independent, monoenergetic, X,Y-geometry neutron transport equation for the forward transport problem and the equation which is adjoint to it for the adjoint transport problem. The well-known reciprocity relation is used to correlate these two problems, yielding an explicit relation between interior sources and the power generated by the fuel regions. The discrete ordinates (SN) formulation is applied to the forward and adjoint transport problems with the level-symmetric angular quadratures for X,Y-geometry calculations. The adjoint SN problems are solved by a new coarse-mesh method named adjoint response matrix-constant nodal (RM†-CN) method which is used with the adjoint partial one-node block inversion iterative scheme to generate numerical values for the adjoint node-edge average angular fluxes. The RM†-CN method is a companion method of the response matrix-constant nodal method used to solve forward SN transport problems. Numerical results are given to verify the offered methodology.

An analytical discrete ordinates nodal solution to the two-dimensional adjoint transport problem

A solution to the one-group discrete ordinates approximation of the adjoint to the transport equation in two-dimensional Cartesian geometry is developed in this work. Starting from an integration process over a region (node) of the domain, the model is reduced to one-dimensional transverse-integrated equations in the x and y variables which are then solved by the Analytical Discrete Ordinates Method (ADO Method). Particular attention is given to the effect of non-homogeneous boundary conditions in the formulation. Explicit expressions are derived and used for evaluating, in a fast way, absorption rates of internal detectors from the adjoint formulation. Excellent agreement was found between numerical results obtained for the absorption rates evaluated from the forward and adjoint problems.

Response Matrix Method–Based Importance Solver and Variance Reduction Scheme in the Serpent 2 Monte Carlo Code

A deterministic importance solver has been implemented as an internal subroutine in the Serpent 2 Monte Carlo code for the purpose of producing weight-window meshes for variance reduction. The routine solves the adjoint transport problem using the response matrix method with coupling coefficients obtained from a conventional forward Monte Carlo simulation. The methodology can be applied to photon and neutron external source problems, and the solver supports multiple energy groups and several mesh types. Importances can be generated with respect to multiple responses, and an iterative global variance reduction sequence enables distributing the transported particle population evenly throughout the geometry. This paper describes the methodology applied in the response matrix solver and presents a verification for the generated importance functions through simple demonstrations. A practical example involving a photon shielding problem is included for performance evaluation.

A numerical method for monoenergetic slab-geometry fixed-source adjoint transport problems in the discrete ordinates formulation with no spatial truncation error

A numerical method that is free of spatial truncation errors is developed for one-speed slab-geometry constant fixed-source adjoint discrete ordinates (SN) problems. The unknowns in the method are the cell-edge and cell-average adjoint angular fluxes, and the numerical values obtained for these quantities are those of the analytic solution of the adjoint SN equations. The method is based on the use of the standard spatially discretised adjoint SN balance equations, which hold in each spatial cell and for each discrete ordinates direction, and a non-standard adjoint auxiliary equation that contains a Green’s function for the cell-average adjoint angular flux in terms of the exiting cell-edge adjoint angular fluxes and the interior adjoint source. Numerical results are given to illustrate the method’s accuracy.

Multigroup Adjoint Transport Solution Using the Method of Cyclic Characteristics

Most perturbation theory calculation methods for neutron transport problems are based on the assumption that the solution to the adjoint transport problem is known. Here we develop an adjoint transport solution based on the method of cyclic characteristics (MOCC) for two-dimensional fuel assembly problems with isotropic scattering. The main advantages of the MOCC method are (a) it requires lower computing time and memory spaces than the collision probability (CP) method and (b) it does not require the boundary surface currents as for the method of characteristics with isotropic tracking. In the MOCC the adjoint characteristics equations associated with a cyclic tracking line are formulated in such a way that a closed form for the adjoint angular function can be obtained. The mathematical relationship between the adjoint function obtained by CP method and the adjoint function by MOCC is also presented. In order to speed up the MOCC solution algorithm, group-reduction and group-splitting techniques based on the structure of the adjoint scattering matrix are implemented. In addition, a combined forward flux/adjoint function iteration scheme, based on the group-splitting technique and the common use of large numbers of variables storing tracking-line data and exponential values, is proposed to reduce the computing time. To demonstrate the efficiency of these algorithms, calculations are performed on a 17 × 17 pressurized water reactor lattice, a 37-pin CANDU [Canada deuterium uranium reactor] cell, and the Watanabe-Maynard benchmark. Comparisons of adjoint function and keff results obtained by the MOCC and the CP method are presented.

A four-group analytic rod transport problem for ( n, γ) and ( n, n′γ) verification

Forward and adjoint transport problems are solved analytically on a homogeneous rod (the forward-backward model) with four energy groups. Groups 1 and 2 represent fast and thermal neutrons; groups 3 and 4 represent gamma rays born from thermal neutron capture and fast neutron inelastic scattering, respectively. Neutron scattering includes self-scattering and downscattering. There is no fission. Gamma rays are treated as lines, so gamma-ray scattering is ignored. Neutron scattering and gamma production are isotropic. In the forward problem, there are neutron sources incoming at the rod endpoints but no internal neutron sources; there are no extrinsic gamma sources. In the adjoint problem, there are gamma detectors at the rod endpoints but no neutron detectors anywhere. PARTISN results agree to seven significant figures with the analytic results in a numerical benchmark problem using hydrogen and carbon. The equivalence among the rod transport problem, diffusion theory, and the S 2 discrete ordinates method is demonstrated for didactic purposes.

Scattering and absorption transport sensitivity functions for optical tomography

Optical tomography is modelled as an inverse problem for the time-dependent linear transport equation. We decompose the linearized residual operator of the problem into absorption and scattering transport sensitivity functions. We show that the adjoint linearized residual operator has a similar physical meaning in optical tomography as the 'backprojection' operator in x-ray tomography. In this interpretation, the geometric patterns onto which the residuals are backprojected are given by the same absorption and scattering transport sensitivity functions which decompose the forward residual operator. Moreover, the 'backtransport' procedure, which has been introduced in an earlier paper by the author, can then be interpreted as an efficient scheme for 'backprojecting' all (filtered) residuals corresponding to one source position simultaneously into the parameter space by just solving one adjoint transport problem. Numerical examples of absorption and scattering transport sensitivity functions for various situations (including applications with voids) are presented.

Industrial pollution transport. Part 2. Control of industrial emissions

Solutions of the pollution transport problem and its adjoint are used to monitor mean pollution concentration in an ecologically important zone. Four strategies of control over pollutants released into the atmosphere by industrial plants are suggested. They differ by the restrictions imposed on the emission rate of each plant. All the strategies use solutions of the adjoint transport problem and assure the fulfillment of the sanitary norm in the zone. A linear interpolation of these strategies also brings pollution level in the zone down to the sanitary norm. A method of detecting the plants violating the prescribed emission rates is also given. A simple example is given to illustrate the strategies suggested.

A transport-backtransport method for optical tomography

Optical tomography is modelled by the inverse problem of the time-dependent linear transport equation in n spatial dimensions (n = 2,3). Based on the measurements which consist of some functionals of the outgoing density at the boundary for different sources , , two coefficients of the equation, the absorption coefficient and the scattering coefficient b(x), are reconstructed simultaneously inside . Starting out from some initial guess for these coefficients, the transport-backtransport (TBT) algorithm calculates the difference between the computed and the physically given measurements for a fixed source by solving a `direct' transport problem, and then transports these residuals back into the medium by solving a corresponding adjoint transport problem. The correction to the guess is calculated from the densities of the direct and the adjoint problem inside the medium. Doing this for all source positions , , one after the other yields one sweep of the algorithm. Numerical experiments are presented for the case when n = 2. They show that the TBT-method is able to reconstruct and to distinguish between scattering and absorbing objects in the case of large mean free path (which corresponds to x-ray tomography with scattering). In the case of very small mean free path (which corresponds to optical tomography), scattering and absorbing objects are located during the early sweeps, but phantoms are built up in the reconstructed scattering coefficient at positions where an absorber is situated and vice versa.

Automatic Variance Reduction for Three-Dimensional Monte Carlo Simulations by the Local Importance Function Transform—I: Analysis

A new automated variance reduction method for the Monte Carlo simulation of multigroup neutron transport source-detector problems is described. The method is based on a modified transport problem that can be solved by analog Monte Carlo with zero variance. The implementation of this modified problem is impractical, in part because it requires the exact solution of an adjoint transport problem. The new local importance function transform (LIFT) method is developed to overcome this difficulty by approximating the exact adjoint solution with a piecewise-continuous function containing parameters that are obtained from a deterministic adjoint calculation. The transport and collision processes of the transformed Monte Carlo problem bias source distribution, distance to collision, and selection of postcollision energy groups and directions. A companion paper provides numerical results that demonstrate the efficiency of the LIFT method.

A general nonlinear inverse transport algorithm using forward and adjoint flux computations

Iterative approaches to the nonlinear inverse transport problem are described, which give rise to the structure that best predicts a set of transport observations. Such methods are based on minimizing a global error functional measuring the discrepancy between predicted and observed transport data. Required for this minimization is the functional gradient (Frechet derivative) of the global error evaluated with respect to a set of unknown material parameters (specifying boundary locations, scattering cross sections, etc.) which are to be determined. It is shown how this functional gradient is obtained from numerical solutions to the forward and adjoint transport problems computed once per iteration. This approach is not only far more efficient, but also more accurate, than a finite-difference method for computing the gradient of the global error. The general technique can be applied to inverse-transport problems of all descriptions, provided only that solutions to the forward and adjoint problems can be found numerically. As an illustration, two inverse problems are treated: the reconstruction of an anisotropic scattering function in a one-dimensional homogeneous slab and the two-dimensional imaging of a spatially-varying scattering cross section.