Self-adjoint Angular Flux Research Articles

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.

A new mathematical adjoint for the modified SAAF-[formula omitted] equations

We present a new adjoint FEM weak form, which can be directly used for evaluating the mathematical adjoint, suitable for perturbation calculations, of the self-adjoint angular flux SN equations (SAAF-SN) without construction and transposition of the underlying coefficient matrix. Stabilization schemes incorporated in the described SAAF-SN method make the mathematical adjoint distinct from the physical adjoint, i.e. the solution of the continuous adjoint equation with SAAF-SN.This weak form is implemented into RattleSnake, the MOOSE (Multiphysics Object-Oriented Simulation Environment) based transport solver.Numerical results verify the correctness of the implementation and show its utility both for fixed source and eigenvalue problems.

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.

Nonlinear Acceleration of a Continuous Finite Element Discretization of the Self-Adjoint Angular Flux Form of the Transport Equation

Nonlinear acceleration of a continuous finite element (CFE) discretization of the transport equation requires a modification of the transport solution in order to achieve local conservation, a condition used in nonlinear acceleration to define the stopping criterion. In this work we implement a coarse-mesh finite difference acceleration for a CFE discretization of the second-order self-adjoint angular flux (SAAF) form of the transport equation and use a postprocessing to enforce local conservation. Numerical results are given for one-group source calculations of one-dimensional slabs. We also give a novel formal derivation of the boundary conditions for the SAAF.

A spherical harmonics—Finite element discretization of the self-adjoint angular flux neutron transport equation

The spherical harmonics (P N) method is widely used in solving the neutron transport equation, but it has some disadvantages. One of them comes from the complexity of the P N equations. Another one comes from the difficulty of dealing with the vacuum boundary condition exactly. In this paper, the P N method is applied to the self-adjoint angular flux (SAAF) neutron transport equation and a set of P N moments equations coupled with each other are obtained. An iterative method is utilized to decouple them and solve them moment by moment. The corresponding vacuum boundary condition is derived based on the Marshak boundary condition. The spatial variables are discretized on unstructured-meshes by use of the finite element method (FEM). The numerical results of several problems demonstrate that this method can provide high precision results and avoid the ray effect, which appears in the discrete ordinate (S N) method, with relatively high computational efficiency.

Streamline upwind Petrov–Galerkin methods for the steady-state Boltzmann transport equation

This paper describes Petrov–Galerkin finite element methods for solving the steady-state Boltzmann transport equation. The methods in their most general form are non-linear and therefore capable of accurately resolving sharp gradients in the solution field. In contrast to previously developed non-linear Petrov–Galerkin (shock-capturing) schemes the methods described in this paper are streamline based. The methods apply a finite element treatment for the internal domain and a Riemann approach on the boundary of the domain. This significantly simplifies the numerical application of the scheme by circumventing the evaluation of complex half-range angular integrals at the boundaries of the domain. The underlying linear Petrov–Galerkin scheme is compared with other Petrov–Galerkin methods found in the radiation transport literature such as those based on the self-adjoint angular flux equation (SAAF) [G.C. Pomraning, Approximate methods of solution of the monoenergetic Boltzmann equation, Ph.D. Thesis, MIT, 1962] and the even-parity (EP) equation [V.S. Vladmirov, Mathematical methods in the one velocity theory of particle transport, Atomic Energy of Canada Limited, Ontario, 1963]. The relationship of the linear method to Riemann methods is also explored.The Petrov–Galerkin methods developed in this paper are applied to a variety of 2-D steady-state and fixed source radiation transport problems. The examples are chosen to cover the range of different radiation regimes from optically thick (diffusive and/or highly absorbing) to transparent and semi-transparent media. These numerical examples show that the non-linear Petrov–Galerkin method is capable of producing accurate, oscillation free solutions across the full spectrum of radiation regimes.

Spatial discretizations for self-adjoint forms of the radiative transfer equations

There are three commonly recognized second-order self-adjoint forms of the neutron transport equation: the even-parity equations, the odd-parity equations, and the self-adjoint angular flux equations. Because all of these equations contain second-order spatial derivatives and are self-adjoint for the mono-energetic case, standard continuous finite-element discretization techniques have proved quite effective when applied to the spatial variables. We first derive analogs of these equations for the case of time-dependent radiative transfer. The primary unknowns for these equations are functions of the angular intensity rather than the angular flux, hence the analog of the self-adjoint angular flux equation is referred to as the self-adjoint angular intensity equation. Then we describe a general, arbitrary-order, continuous spatial finite-element approach that is applied to each of the three equations in conjunction with backward-Euler differencing in time. We refer to it as the “standard” technique. We also introduce an alternative spatial discretization scheme for the self-adjoint angular intensity equation that requires far fewer unknowns than the standard method, but appears to give comparable accuracy. Computational results are given that demonstrate the validity of both of these discretization schemes.

Finite Element Solution of the Self-Adjoint Angular Flux Equation for Coupled Electron-Photon Transport

A novel approach is proposed for charged particle transport calculations using a recently developed second-order, self-adjoint angular flux (SAAF) form of the Boltzmann transport equation with continuous slowing-down. A finite element discretization that is linear continuous in space and linear discontinuous (LD) in energy is described and implemented in a one-dimensional, planar geometry, multigroup, discrete ordinates code for charged particle transport. The cross-section generating code CEPXS is used to generate the electron and photon transport cross sections employed in this code. The discrete ordinates SAAF transport equation is solved using source iteration in conjunction with an inner iteration acceleration scheme and an outer iteration acceleration scheme. Outer iterations are required with the LD energy discretization scheme because the two angular flux unknowns within each group are coupled, which gives rise to effective upscattering. The inner iteration convergence is accelerated using diffusion synthetic acceleration, and the outer iteration convergence is accelerated using a diamond difference approximation to the LD energy discretization. Computational results are given that demonstrate the effectiveness of our convergence acceleration schemes and the accuracy of our discretized SAAF equation.