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Performance Analysis of Spatial Discretization Error Estimators for Radiation Transport Discrete Ordinates Methods.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Performance Analysis of Spatial Discretization Error Estimators for Radiation Transport Discrete Ordinates Methods./
Author:	Hart, Nathan Henry.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2020,
Description:	226 p.
Notes:	Source: Dissertations Abstracts International, Volume: 81-11.
Contained By:	Dissertations Abstracts International81-11.
Subject:	Nuclear engineering. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28004324
ISBN:	9798643183358

Performance Analysis of Spatial Discretization Error Estimators for Radiation Transport Discrete Ordinates Methods.
Hart, Nathan Henry.

Performance Analysis of Spatial Discretization Error Estimators for Radiation Transport Discrete Ordinates Methods. - Ann Arbor : ProQuest Dissertations & Theses, 2020 - 226 p.

Source: Dissertations Abstracts International, Volume: 81-11.

Thesis (Ph.D.)--North Carolina State University, 2020.

This item must not be sold to any third party vendors.

The solution to the steady-state discrete ordinates transport equation describes the distribution of particles in space, energy, and angle for a given configuration. Despite many layers of approximation, the spatial continuum form of this equation cannot be solved analytically, except for a few idealized cases. Thus, a spatial discretization scheme in the form of a spatial mesh and a discrete representation of the solution is applied to the equation, permitting a numerical solution to be obtained. As the mesh size approaches zero and the discretization scheme order (the order at which the finite-dimension solution representation is truncated) approaches infinity, the numerical solution is expected to approach the true solution. However, short of this limit, the numerical solution departs from the true solution, the solution to the spatially undiscretized discrete ordinates tranport equation, by a quantity defined as the spatial discretization error. As the spatial discretization error represents an inaccuracy in the numerical solution, quantifying and/or minimizing its magnitude and spatial distribution is desirable. Because lack of knowledge of the true solution precludes knowledge of the true spatial discretization error, and the former is generally unobtainable, this process reduces to an approximate estimation of the spatial discretization error. To this end, a novel implicit, residual-based spatial discretization error estimator, the \\residual source estimator", has been derived. Careful definition of the residual leads to a special auxiliary error transport equation, which is analogous to the discretized transport equation and solved with the same methods and on the same discretization space as the original numerical solution. The fixed source in the error transport equation is the residual, and the solution is the spatial discretization error. This formulation is advantageous because, if the true residual is used as fixed source in solving the error transport equation, the exact distribution of the spatial discretization error results; hence accurate representation of the residual is desired, because it is expected to yield an accurate estimate of the error distribution. By locally representing the true solution as a truncated Taylor series expansion, approximations of the residual can be obtained for any discretization scheme as a function of cell size, physical parameters, and pointwise derivatives of the solution. Because the pointwise derivatives that appear in the expression of the approximate residual are outside the space of the numerical solution, they also demand approximation in terms of the numerical solution, thus making the estimator a posteriori. The residual source estimator was previously formulated, implemented, and tested for the Discontinuous Galerkin Finite Element Method of order 0 (DGFEM-0), a piecewise constant scheme, on a suite of Method of Manufactured Solutions (MMS) problems with solutions of varying regularity order and a wide range of physical parameters. These tests illustrated the method's ability to provide a precise and accurate error estimate at a fraction of the computational cost of a comparable h-refinement scheme. This work continues the exploration of the residual source estimator. The residual approximations for Discontinuous Galerkin Finite Element Method of order 1 (DGFEM-1), a piecewise linear scheme, and the Diamond Difference method, as an example of weighted difference schemes, are derived. Furthermore, two methods of derivative approximations are presented. The first is a patch recovery scheme wherein the numerical solution in a "patch" of cells surrounding the node where the pointwise derivative is evaluated is used to build a derivative approximation using the Taylor expansion representation of the solution moments obtained with the DGFEM-Lambda scheme. However, this procedure has been shown to fail for DGFEM-Lambda when the mesh is nonuniform, so a second scheme is derived as an evaluation of the weak derivatives in the finite element space. The residual source estimator for DGFEM-1 is compared to an h-refinement error estimator and a heuristically-derived local error indicator for three cases of varying true solution behavior. These results show that the residual source estimator uniquely suffers greatly from insufficiently accurate numerical solutions in the presence of prominent singular characteristics (SCs). Following the assessment of error estimation for DGFEM-1, a special treatment for DGFEM-0 for cells intersected by the SCs is examined for the purpose of greatly reducing the error in the numerical solution inflicted by solution irregularity in the vicinity of the SCs. This is accomplished by implementing the singular characteristic tracking scheme, in which a cell traversed by the SC is split into two subregions on-the-fly, above and below the SC trajectory within the cell, and the step characteristics method is used to obtain the solution separately in each subregion. Separate out ow values are computed and retained to use for computing the downwind cells in the sweep that are also intersected by the SC. By only treating the numerical solution as smooth where the true solution is smooth (for a given ordinate), the numerical solution provides a superior representation of the true solution because of the substantial reduction in error incurred in the discretized streaming term due to true solution irregularities within the cell. For computational efficiency, this procedure is only performed once as an additional inner iteration after the iterative stopping condition is satisfied by the iterative solution. For problems where the discontinuity in the solution is large in magnitude and unattenuated by the materials, this partial singular characteristic tracking scheme produces superior residual approximations in the plume of reduced-accuracy numerical solution surrounding the SCs, which are induced by the spread of error in the particle transport problem. Furthermore, the residual approximation, which has undergone no additional enhancement from the standard method, is accurate in the SC-intersected cells, where the residual and derivative approximations are known to be incorrect, indicating some cancellation of error. These two effect lead to a more accurate error estimate. Finally, the error estimators are tested on two realistic problem geometries, a C5G7 UO2 assembly (fast and thermal groups) and a dogleg duct shielding configuration. For these problem configurations involving multiple material heterogeneities, the true solution is irregular along SCs, of which there are many. Therefore, estimator performance is highly dependent on attenuation of the magnitude of irregularity across the SCs, which in turn is dependent on the total cross section of the materials. The residual source estimator performs well under these conditions but has some shortcomings, particularly in a DGFEM-1 discretization scheme in a problem with optically thin material. However, it is found that the two refinement-based estimators considered in this study are biased favorably by the ideal MMS cases. Because on either side of the SC for a given discrete ordinate, either the x- or y- derivative (and all associated cross derivatives) are nil for our MMS, a refinement of a DGFEM-1 solution will produce an especially accurate representation of the true solution. Therefore, they suffer a deterioration in performance for DGFEM-1 in realistic problems versus their performance on the MMS configurations, both being less accurate and precise. However, the general trends of estimator behavior observed in the MMS suite hold.

ISBN: 9798643183358Subjects--Topical Terms:

595435
Nuclear engineering.
Subjects--Index Terms:

Performance analysis

Performance Analysis of Spatial Discretization Error Estimators for Radiation Transport Discrete Ordinates Methods.
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245 1 0 $a Performance Analysis of Spatial Discretization Error Estimators for Radiation Transport Discrete Ordinates Methods.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2020
300 $a 226 p.
500 $a Source: Dissertations Abstracts International, Volume: 81-11.
500 $a Advisor: Azmy, Yousry Y.
502 $a Thesis (Ph.D.)--North Carolina State University, 2020.
506 $a This item must not be sold to any third party vendors.
520 $a The solution to the steady-state discrete ordinates transport equation describes the distribution of particles in space, energy, and angle for a given configuration. Despite many layers of approximation, the spatial continuum form of this equation cannot be solved analytically, except for a few idealized cases. Thus, a spatial discretization scheme in the form of a spatial mesh and a discrete representation of the solution is applied to the equation, permitting a numerical solution to be obtained. As the mesh size approaches zero and the discretization scheme order (the order at which the finite-dimension solution representation is truncated) approaches infinity, the numerical solution is expected to approach the true solution. However, short of this limit, the numerical solution departs from the true solution, the solution to the spatially undiscretized discrete ordinates tranport equation, by a quantity defined as the spatial discretization error. As the spatial discretization error represents an inaccuracy in the numerical solution, quantifying and/or minimizing its magnitude and spatial distribution is desirable. Because lack of knowledge of the true solution precludes knowledge of the true spatial discretization error, and the former is generally unobtainable, this process reduces to an approximate estimation of the spatial discretization error. To this end, a novel implicit, residual-based spatial discretization error estimator, the \\residual source estimator", has been derived. Careful definition of the residual leads to a special auxiliary error transport equation, which is analogous to the discretized transport equation and solved with the same methods and on the same discretization space as the original numerical solution. The fixed source in the error transport equation is the residual, and the solution is the spatial discretization error. This formulation is advantageous because, if the true residual is used as fixed source in solving the error transport equation, the exact distribution of the spatial discretization error results; hence accurate representation of the residual is desired, because it is expected to yield an accurate estimate of the error distribution. By locally representing the true solution as a truncated Taylor series expansion, approximations of the residual can be obtained for any discretization scheme as a function of cell size, physical parameters, and pointwise derivatives of the solution. Because the pointwise derivatives that appear in the expression of the approximate residual are outside the space of the numerical solution, they also demand approximation in terms of the numerical solution, thus making the estimator a posteriori. The residual source estimator was previously formulated, implemented, and tested for the Discontinuous Galerkin Finite Element Method of order 0 (DGFEM-0), a piecewise constant scheme, on a suite of Method of Manufactured Solutions (MMS) problems with solutions of varying regularity order and a wide range of physical parameters. These tests illustrated the method's ability to provide a precise and accurate error estimate at a fraction of the computational cost of a comparable h-refinement scheme. This work continues the exploration of the residual source estimator. The residual approximations for Discontinuous Galerkin Finite Element Method of order 1 (DGFEM-1), a piecewise linear scheme, and the Diamond Difference method, as an example of weighted difference schemes, are derived. Furthermore, two methods of derivative approximations are presented. The first is a patch recovery scheme wherein the numerical solution in a "patch" of cells surrounding the node where the pointwise derivative is evaluated is used to build a derivative approximation using the Taylor expansion representation of the solution moments obtained with the DGFEM-Lambda scheme. However, this procedure has been shown to fail for DGFEM-Lambda when the mesh is nonuniform, so a second scheme is derived as an evaluation of the weak derivatives in the finite element space. The residual source estimator for DGFEM-1 is compared to an h-refinement error estimator and a heuristically-derived local error indicator for three cases of varying true solution behavior. These results show that the residual source estimator uniquely suffers greatly from insufficiently accurate numerical solutions in the presence of prominent singular characteristics (SCs). Following the assessment of error estimation for DGFEM-1, a special treatment for DGFEM-0 for cells intersected by the SCs is examined for the purpose of greatly reducing the error in the numerical solution inflicted by solution irregularity in the vicinity of the SCs. This is accomplished by implementing the singular characteristic tracking scheme, in which a cell traversed by the SC is split into two subregions on-the-fly, above and below the SC trajectory within the cell, and the step characteristics method is used to obtain the solution separately in each subregion. Separate out ow values are computed and retained to use for computing the downwind cells in the sweep that are also intersected by the SC. By only treating the numerical solution as smooth where the true solution is smooth (for a given ordinate), the numerical solution provides a superior representation of the true solution because of the substantial reduction in error incurred in the discretized streaming term due to true solution irregularities within the cell. For computational efficiency, this procedure is only performed once as an additional inner iteration after the iterative stopping condition is satisfied by the iterative solution. For problems where the discontinuity in the solution is large in magnitude and unattenuated by the materials, this partial singular characteristic tracking scheme produces superior residual approximations in the plume of reduced-accuracy numerical solution surrounding the SCs, which are induced by the spread of error in the particle transport problem. Furthermore, the residual approximation, which has undergone no additional enhancement from the standard method, is accurate in the SC-intersected cells, where the residual and derivative approximations are known to be incorrect, indicating some cancellation of error. These two effect lead to a more accurate error estimate. Finally, the error estimators are tested on two realistic problem geometries, a C5G7 UO2 assembly (fast and thermal groups) and a dogleg duct shielding configuration. For these problem configurations involving multiple material heterogeneities, the true solution is irregular along SCs, of which there are many. Therefore, estimator performance is highly dependent on attenuation of the magnitude of irregularity across the SCs, which in turn is dependent on the total cross section of the materials. The residual source estimator performs well under these conditions but has some shortcomings, particularly in a DGFEM-1 discretization scheme in a problem with optically thin material. However, it is found that the two refinement-based estimators considered in this study are biased favorably by the ideal MMS cases. Because on either side of the SC for a given discrete ordinate, either the x- or y- derivative (and all associated cross derivatives) are nil for our MMS, a refinement of a DGFEM-1 solution will produce an especially accurate representation of the true solution. Therefore, they suffer a deterioration in performance for DGFEM-1 in realistic problems versus their performance on the MMS configurations, both being less accurate and precise. However, the general trends of estimator behavior observed in the MMS suite hold.
590 $a School code: 0155.
650 4 $a Nuclear engineering. $3 595435
653 $a Performance analysis
653 $a Spatial discretization error estimators
653 $a Radiation transport discrete ordinates methods
690 $a 0552
710 2 $a North Carolina State University. $3 1018772
773 0 $t Dissertations Abstracts International $g 81-11.
790 $a 0155
791 $a Ph.D.
792 $a 2020
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28004324