Diffusion Acceleration Schemes for Self-Adjoint Angular Flux Formulation with a Void Treatment

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Second order forms of the transport equation allow the use of continuous finite elements (CFEM). This can be desired in multi-physics calculations where other physics require CFEM discretizations. Secondorder transport operators are generally self-adjoint, yielding symmetric positive-definite matrices, which allow the use of efficient linear algebra solvers with an enormous advantage in memory usage.Least-squares (LS) forms of the transport equation can circumvent the void problems of other second order forms, but are almost always non-conservative. Additionally, the standard LS form is not compatible with discrete ordinates method (SN ) iterative solution techniques such as source iteration. A new form of the least-squares transport equation has recently been developed that is compatible with voids and standard SN iterative solution techniques. Performing Nonlinear Diffusion Acceleration (NDA) using an independently-differenced low-order equation enforces conservation for the whole system, and makes this equation suitable for reactor physics calculations. In this context independent means that both the transport and low-order solutions converge to the same scalar flux and current as the spatial mesh is refined, but for a given mesh, the solutions are not necessarily equal.In this paper we show that introducing a weight function to this least-squares equation improves issues with causality and can render our equation equal to the Self-Adjoint Angular Flux (SAAF) equation. Causality is a principle of the transport equation which states that information only travels downstream along characteristics. This principle can be violated numerically. We show how to limit the weight function in voids and demonstrate the effect of this limit on the accuracy. Using the C5G7 benchmark, we compare our method to the self-adjoint angular flux formulation with a void treatment (SAAFτ ), which is not self-adjoint and has a non-symmetric coefficient matrix. We show that the weighted leastsquares equation with NDA gives acceptable accuracy relative to the SAAFτ equation while maintaining a symmetric positive-definite system matrix.

Section: Iia Self-adjoint Angular Flux Equation With Void Treatmentmentioning

confidence: 99%

A Weighted Least-Squares Transport Equation Compatible with Source Iteration and Voids

Hammer

Morel

Wang

2018

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“…This is in particular the case of the LS method compatible with voids [15] and can significantly degrade the accuracy of the solution, in particular for k-eigenvalue problems [16]. Acceleration schemes such as Nonlinear Diffusion Acceleration (NDA) can help recover global conservation [12] but such schemes have yet to be developed for P N methods.…”

Section: Introductionmentioning

confidence: 99%

Globally Conservative, Hybrid Self-Adjoint Angular Flux and Least-Squares Method Compatible with Voids

Labouré

McClarren

Wang

2017

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In this paper, we derive a method for the second-order form of the transport equation that is both globally conservative and compatible with voids, using Continuous Finite Element Methods (CFEM). The main idea is to use the Least-Squares (LS) form of the transport equation in the void regions and the Self-Adjoint Angular Flux (SAAF) form elsewhere. While the SAAF formulation is globally conservative, the LS formulation need a correction in void. The price to pay for this fix is the loss of symmetry of the bilinear form. We first derive this Conservative LS (CLS) formulation in void. Second we combine the SAAF and CLS forms and end up with an hybrid SAAF-CLS method, having the desired properties. We show that extending the theory to near-void regions is a minor complication and can be done without affecting the global conservation of the scheme. Being angular discretization agnostic, this method can be applied to both discrete ordinates (S N ) and spherical harmonics (P N ) methods. However, since a globally conservative and void compatible second-order form already exists for S N (Wang et al. 2014), but is believed to be new for P N , we focus most of our attention on that latter angular discretization. We implement and test our method in Rattlesnake within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework. Results comparing it to other methods are presented.

“…Symmetric successive overrelaxation [27] is used as preconditioner for all calculations with relaxation factor fixed at 1.4. In all tests, we also include results from solving the globally conservative selfadjoint angular flux (SAAF) equation with CFEM as a comparison [2,21]. In all problems, piecewise linear polynomial basis functions are used.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…CFEM-SAAF is globally conservative, yet, not compatible with void and potentially ill-conditioned in near-void situations. Therefore, efforts have been put in alleviating CFEM-SAAF in void [17,18,21,24]. We will specially give a brief review of the treatment developed in [18,24].…”

Section: Conservative Void Treatmentmentioning

confidence: 99%

An Accurate Globally Conservative Subdomain Discontinuous Least-Squares Scheme for Solving Neutron Transport Problems

Zheng

McClarren

Morel

2017

In this work, we present a subdomain discontinuous least-squares (SDLS) scheme for neutronics problems. Least-squares (LS) methods are known to be inaccurate for problems with sharp total-cross section interfaces. In addition, the least-squares scheme is known not to be globally conservative in heterogeneous problems. In problems where global conservation is important, e.g. k-eigenvalue problems, a conservative treatment must be applied. We, in this study, propose an SDLS method that retains global conservation, and, as a result, gives high accuracy on eigenvalue problems. Such a method resembles the LS formulation in each subdomain without a material interface and differs from LS in that an additional least-squares interface term appears for each interface. The scalar flux is continuous in each subdomain with continuous finite element method (CFEM) while discontinuous on interfaces for every pair of contiguous subdomains. SDLS numerical results are compared with those obtained from other numerical methods with test problems having material interfaces. High accuracy of scalar flux in fixed-source problems and k eff in eigenvalue problems are demonstrated.