Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

Mirzargar, Mahsa; Jallepalli, Ashok; Ryan, Jennifer K.; Kirby, Robert M.

doi:10.1007/s10915-017-0517-5

Cited by 9 publications

(7 citation statements)

References 32 publications

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“…In [51,58,71], variations of the SIAC filter, called one-sided filters, are introduced to deal with boundaries and mathematical discontinuities in the solution (e.g., shocks in supersonic compressible flows). To improve accuracy on hexagonal meshes, Mirzargar et al [49] proposed hexagonal SIAC filters. Li et al [43] discussed effective ways to calculate the derivative quantities using SIAC and one-sided SIAC filters.…”

Section: Data Transformation Methodologiesmentioning

confidence: 99%

The Effect of Data Transformations on Scalar Field Topological Analysis of High-Order FEM Solutions

Jallepalli

Levine

Kirby

2020

IEEE Trans. Visual. Comput. Graphics

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a) Sampled vorticity. (b) Subdivided vorticity.(c) L-SIAC vorticity. Fig. 1: Topological segmentation of counter-rotating vortex sampled using different methodologies discussed in the paper and by filtering the contour tree for segments that resemble vortex-like structures. The number, shape, and boundaries of the segments are different for the three techniques.Abstract-High-order finite element methods (HO-FEM) are gaining popularity in the simulation community due to their success in solving complex flow dynamics. There is an increasing need to analyze the data produced as output by these simulations.Simultaneously, topological analysis tools are emerging as powerful methods for investigating simulation data. However, most of the current approaches to topological analysis have had limited application to HO-FEM simulation data for two reasons. First, the current topological tools are designed for linear data (polynomial degree one), but the polynomial degree of the data output by these simulations is typically higher (routinely up to polynomial degree six). Second, the simulation data and derived quantities of the simulation data have discontinuities at element boundaries, and these discontinuities do not match the input requirements for the topological tools. One solution to both issues is to transform the high-order data to achieve low-order, continuous inputs for topological analysis. Nevertheless, there has been little work evaluating the possible transformation choices and their downstream effect on the topological analysis. We perform an empirical study to evaluate two commonly used data transformation methodologies along with the recently introduced L-SIAC filter for processing high-order simulation data. Our results show diverse behaviors are possible. We offer some guidance about how best to consider a pipeline of topological analysis of HO-FEM simulations with the currently available implementations of topological analysis.

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Section: Data Transformation Methodologiesmentioning

confidence: 99%

The Effect of Data Transformations on Scalar Field Topological Analysis of High-Order FEM Solutions

Jallepalli

Levine

Kirby

2020

IEEE Trans. Visual. Comput. Graphics

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“…Although the discussion is limited to one-dimension, it can be extended to Cartesian meshes in more than one space dimension using a tensor product approach. More advanced applications of the multi-dimensional SIAC post-processor are the Hexagonal SIAC [22] or Line SIAC [11].…”

Section: Siac Post-processorsmentioning

confidence: 99%

Residual Estimates for Post-processors in Elliptic Problems

et al. 2021

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In this work we examine a posteriori error control for post-processed approximations to elliptic boundary value problems. We introduce a class of post-processing operator that “tweaks” a wide variety of existing post-processing techniques to enable efficient and reliable a posteriori bounds to be proven. This ultimately results in optimal error control for all manner of reconstruction operators, including those that superconverge. We showcase our results by applying them to two classes of very popular reconstruction operators, the Smoothness-Increasing Accuracy-Conserving filter and superconvergent patch recovery. Extensive numerical tests are conducted that confirm our analytic findings.

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“…x H . A computationally efficient alternative to the tensor product case is to use the Hexagonal SIAC filter (HSIAC) by Mirzarger et al [13], or the Line SIAC filter introduced by Docampo et al [6] and applied to problems in visualization problems by Jallepalli et al [9].…”

Section: Siac Filtermentioning

confidence: 99%

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Discontinuous Galerkin Solutions over Nonuniform Meshes: Superconvergence and Optimal Accuracy

Ryan

Kirby

et al. 2019

J Sci Comput

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Smoothness-increasing accuracy-conserving (SIAC) filtering is an area of increasing interest because it can extract the "hidden accuracy" in discontinuous Galerkin (DG) solutions. It has been shown that by applying a SIAC filter to a DG solution, the accuracy order of the DG solution improves from order k + 1 to order 2k + 1 for linear hyperbolic equations over uniform meshes. However, applying a SIAC filter over nonuniform meshes is difficult, and the quality of filtered solutions is usually unsatisfactory applied to approximations defined on nonuniform meshes. The applicability to such approximations over nonuniform meshes is the biggest obstacle to the development of a SIAC filter. The purpose of this paper is twofold: to study the connection between the error of the filtered solution and the nonuniform mesh and to develop a filter scaling that approximates the optimal error reduction. First, through analyzing the error estimates for SIAC filtering, we computationally establish for the first time a relation between the filtered solutions and the unstructuredness of nonuniform meshes. Further, we demonstrate that there exists an optimal accuracy of the filtered solution for a given nonuniform mesh and that it is possible to obtain this optimal accuracy by the method we propose, an optimal filter scaling. By applying the newly designed filter scaling over nonuniform meshes, the filtered solution has demonstrated improvement in accuracy order as well as reducing the error compared to the original DG solution. Finally, we apply the proposed methods over a large number of nonuniform meshes and compare the performance with existing methods to demonstrate the superiority of our method.

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Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

Cited by 9 publications

References 32 publications

The Effect of Data Transformations on Scalar Field Topological Analysis of High-Order FEM Solutions

The Effect of Data Transformations on Scalar Field Topological Analysis of High-Order FEM Solutions

Residual Estimates for Post-processors in Elliptic Problems

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Discontinuous Galerkin Solutions over Nonuniform Meshes: Superconvergence and Optimal Accuracy

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