Treatment of voids in finite element transport methods

Ackroyd, R.T.; Issa, J.G.; Riyait, N.S.

doi:10.1016/0149-1970(86)90015-6

Cited by 19 publications

(11 citation statements)

References 4 publications

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“…The problem of voids has been the subject of considerable research in the particle transport literature. [25][26][27][28] A good summary is given by Ackroyd, 13 Chap. 10.…”

Section: The Void Problemmentioning

confidence: 99%

The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions

et al. 2000

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We present a method for handling nonscattering regions within diffusing domains. The method develops from an iterative radiosity-diffusion approach using Green's functions that was computationally slow. Here we present an improved implementation using a finite element method (FEM) that is direct. The fundamental idea is to introduce extra equations into the standard diffusion FEM to represent nondiffusive light propagation across a nonscattering region. By appropriate mesh node ordering the computational time is not much greater than for diffusion alone. We compare results from this method with those from a discrete ordinate transport code, and with Monte Carlo calculations. The agreement is very good, and, in addition, our scheme allows us to easily model time-dependent and frequency domain problems.

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“…The problem of voids has been the subject of considerable research in the particle transport literature. [25][26][27][28] A good summary is given by Ackroyd, 13 Chap. 10.…”

Section: The Void Problemmentioning

confidence: 99%

The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions

et al. 2000

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“…Current developments in modeling and simulation raised the needs for tools which are able to handle voids or near voids. While this is definitely possible with the first order transport equation, second order schemes often show singularities and conditioning or convergence problems for very small (near zero) total cross sections [6,7]. Certain least-squares (LS) forms of the transport equation can circumvent the void problems of other second order forms, but are non-conservative, which explains why they are not commonly used in the nuclear community.…”

Section: Introductionmentioning

confidence: 99%

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

Hammer

Morel

Wang

2018

Nuclear Science and Engineering

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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.

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“…Perhaps more important, they experience numerical difficulties in near-voids (Ackroyd, Issa, and Riyait, 1986). Here we derive a new form of second-order self-adjoint transport equation that has the advantage relative to standard forms that it can be used in voids or near voids.…”

mentioning

confidence: 99%

A Least-Squares Transport Equation Compatible with Voids

Peterson

Morel

Ragusa

et al. 2014

Journal of Computational and Theoretical Transport

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A L e a s t-S q u a re s T ra n s p o rt E q u a tio n C o m p a t ib le w ith V oidsStandard second-order self-adjoint forms of the transport equation, such as the evenparity, odd-parity, and self-adjoint angular flux equation, cannot be used in voids. Per haps 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 equa tion with both equations being applicable in a void and having a nonconservative ana lytic 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 ID slab-geometry version of our scheme with respect to void solu tions, 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.

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Treatment of voids in finite element transport methods

Cited by 19 publications

References 4 publications

The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions

The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions

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

A Least-Squares Transport Equation Compatible with Voids

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