Linear and quadratic octahedral wavelets on the sphere for angular discretisations of the Boltzmann transport equation

This article presents the first Reduced Order Model (ROM) that efficiently resolves the angular dimension of the time independent, mono-energetic Boltzmann Transport Equation (BTE). It is based on Proper Orthogonal Decomposition (POD) and uses the method of snapshots to form optimal basis functions for resolving the direction of particle travel in neutron/photon transport problems. A unique element of this work is that the snapshots are formed from the vector of angular coefficients relating to a high resolution expansion of the BTE's angular dimension. In addition, the individual snapshots are not recorded through time, as in standard POD, but instead they are recorded through space. In essence this work swaps the roles of the dimensions space and time in standard POD methods, with angle and space respectively. It is shown here how the POD model can be formed from the POD basis functions in a highly efficient manner. The model is then applied to two radiation problems; one involving the transport of radiation through a shield and the other through an infinite array of pins. Both problems are selected for their complex angular flux solutions in order to provide an appropriate demonstration of the model's capabilities. It is shown that the POD model can resolve these fluxes efficiently and accurately. In comparison to high resolution models this POD model can reduce the size of a problem by up to two orders of magnitude without compromising accuracy. Solving times are also reduced by similar factors.

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“…The term S denoted the angularly discretised source, which is a vector of length n m . For full details in computing these terms see [32].…”

Section: The Angular Discretised Transport Equationmentioning

confidence: 99%

A POD reduced order model for resolving angular direction in neutron/photon transport problems

Buchan

Calloo

Goffin

et al. 2015

Journal of Computational Physics

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“…This can be rewritten in a more convenient way in terms of its residual

scriptR (bold-italicΨ) MathClass-rel= (bold-italicA bold-italic \cdot bold-italic \nabla MathClass-bin+ H) bold-italicΨ MathClass-bin- bold-italicS MathClass-rel= 0 MathClass-punc,

where A is the vector of matrices A = ( A x , A y , A z ). A full derivation of this equation is given in .…”

Section: The Sub‐grid Scale Spatial Discretisation Of the Angular Dismentioning

confidence: 99%

A sub‐grid scale finite element agglomeration multigrid method with application to the Boltzmann transport equation

Buchan

Pain

Umpleby

et al. 2012

Numerical Meth Engineering

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SUMMARY This article describes a new element agglomeration multigrid method for solving partial differential equations discretised through a sub‐grid scale finite element formulation. The sub‐grid scale discretisation resolves solution variables through their separate coarse and fine scales, and these are mapped between the multigrid levels using a dual set of transfer operators. The sub‐grid scale multigrid method forms coarse linear systems, possessing the same sub‐grid scale structure as the original discretisation, that can be resolved without them being stored in memory. This is necessary for the application of this article in resolving the Boltzmann transport equation as the linear systems become extremely large. The novelty of this article is therefore a matrix‐free multigrid scheme that is integrated within its own sub‐grid scale discretisation using dual transfer operators and applied to the Boltzmann transport equation. The numerical examples presented are designed to show the method's preconditioning capabilities for a Krylov space‐based solver. The problems range in difficulty, geometry and discretisation type, and comparisons made with established methods show this new approach to perform consistently well. Smoothing operators are also analysed and this includes using the generalized minimal residual method. Here, it is shown that an adaptation to the preconditioned Krylov space is necessary for it to work efficiently. Copyright © 2012 John Wiley & Sons, Ltd.

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“…This work was related to adaptivity and will be discussed in the next paragraph. Research into the two-dimensional wavelet expansion was extended by Buchan et al [5]. In this work, a wavelet basis is constructed on the surface of an octahedron and a linear mapping is used to project this onto the surface of the sphere.…”

Section: Introductionmentioning

confidence: 99%

“…It was noted that the quadratic basis provided little benefit over the linear basis. Buchan further investigated this idea and constructed two other wavelet bases by construction on the surface of a hexahedron [4,6]. Again, the solutions were of comparable accuracy to the standard angular discretisations.…”

Section: Introductionmentioning

confidence: 99%

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Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation

Goffin

Buchan

Dargaville

et al. 2015

Journal of Computational Physics

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Linear and quadratic octahedral wavelets on the sphere for angular discretisations of the Boltzmann transport equation

Cited by 50 publications

References 12 publications

A POD reduced order model for resolving angular direction in neutron/photon transport problems

A POD reduced order model for resolving angular direction in neutron/photon transport problems

A sub‐grid scale finite element agglomeration multigrid method with application to the Boltzmann transport equation

Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation

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