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arxiv: 1907.07224 · v1 · pith:KHRFEWWZnew · submitted 2019-07-16 · 💻 cs.CE · cs.GR

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

Ashok Jallepalli , Joshua A. Levine , Robert M. Kirby This is my paper

Pith reviewed 2026-05-24 20:23 UTC · model grok-4.3

classification 💻 cs.CE cs.GR
keywords high-order FEMtopological analysisdata transformationscalar fieldsL-SIAC filterfinite element simulationsdiscontinuous dataanalysis pipelines
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The pith

The choice of transformation method for high-order FEM data changes the topological features extracted from scalar fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests how converting high-order finite element simulation outputs into continuous low-order data affects later topological analysis. It applies two standard transformation methods plus the L-SIAC filter to simulation results that contain element-boundary discontinuities and measures the resulting differences in extracted features. A reader would care because topological tools are now used to inspect complex flow simulations, yet those tools require transformed inputs whose side effects had not been compared side by side. The study finds that the same underlying data can yield visibly different topological structures depending on the chosen conversion step, so analysis pipelines must select the transformation deliberately rather than by default.

Core claim

The central claim is that transforming high-order FEM simulation data using different methodologies produces diverse behaviors in topological analysis. The empirical comparison of two commonly used transformation methodologies together with the L-SIAC filter shows that each produces distinct topological features from the same high-order scalar fields, and therefore pipelines built with current topological tools require explicit consideration of the transformation choice.

What carries the argument

Data transformation methodologies that convert discontinuous high-order polynomial data into continuous low-order representations for input to topological analysis tools.

If this is right

  • Topological features extracted from HO-FEM scalar fields vary with the transformation method applied before analysis.
  • The L-SIAC filter produces results that differ from the two standard transformation approaches in at least some test cases.
  • Analysis pipelines must evaluate the effect of the chosen transformation on the specific features of interest.
  • Inconsistent topological structures can appear from the same simulation data when different preparation steps are used.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Directly extending topological tools to accept high-order discontinuous inputs would remove the transformation step and its observed variability.
  • Quantifying how each transformation preserves or distorts specific topological invariants could guide method selection for particular analysis goals.
  • Applying the same comparison to vector or tensor fields from the same simulations would test whether the sensitivity is limited to scalar cases.

Load-bearing premise

The two commonly used transformation methodologies together with the L-SIAC filter are representative of possible transformations and the diverse behaviors will generalize to other high-order FEM datasets and topological tools.

What would settle it

Running the same topological extraction on a high-order FEM dataset whose analytic topology is known in advance and finding identical features after every transformation method would falsify the claim of diverse behaviors.

Figures

Figures reproduced from arXiv: 1907.07224 by Ashok Jallepalli, Joshua A. Levine, Robert M. Kirby.

Figure 1
Figure 1. Figure 1: Topological segmentation of counter-rotating vortex sampled using different methodologies discussed in the paper and by [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: The persistence curve, the persistence diagram, and the segmen [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Persistence curves of the sampled data (at three resolutions) as [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 2
Figure 2. Figure 2: The function f(x, y), simulation mesh, and the projected data. To create a higher order representation of f(x, y), we project it on an unstructured triangular element mesh (Figure 2b) over the domain [0,1] × [0,1] with degree 2 polynomial expansions on each element (hereafter referred to as the simulation mesh). With this simple example, we can mimic the elementwise discontinuities we get when computing de… view at source ↗
Figure 6
Figure 6. Figure 6: Persistence curves of the subdivided data (at three resolutions) [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualizations of the persistence diagrams ((a), (b), and (c)) and [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Visualizations of the persistence diagrams ((a), (b), and (c)) and [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Visualizations of the persistence diagrams ((a), (b), and (c)) and [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: Persistence curves of the L-SIAC data (at three resolutions) as [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) Simulation mesh of the flow over a 2D cylinder. (b) Vorticity [PITH_FULL_IMAGE:figures/full_fig_p006_10.png] view at source ↗
Figure 13
Figure 13. Figure 13: The persistence diagrams of the vorticity for the fluid flow past [PITH_FULL_IMAGE:figures/full_fig_p007_13.png] view at source ↗
Figure 11
Figure 11. Figure 11: Persistence curves for the sampled, subdivided, and L-SIAC [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
Figure 14
Figure 14. Figure 14: Segmentation of the vorticity over a flow past a cylinder described [PITH_FULL_IMAGE:figures/full_fig_p007_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Segmentation of the vorticity for the flow past a cylinder de [PITH_FULL_IMAGE:figures/full_fig_p008_15.png] view at source ↗
Figure 18
Figure 18. Figure 18: Zoom in on the region associated with Figure 17. For each of [PITH_FULL_IMAGE:figures/full_fig_p008_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Persistence diagrams for the vorticity of the counter-rotating [PITH_FULL_IMAGE:figures/full_fig_p008_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Segmentation corresponding to the topologically simplified vor [PITH_FULL_IMAGE:figures/full_fig_p009_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Persistence diagrams for the vorticity of the counter-rotating [PITH_FULL_IMAGE:figures/full_fig_p009_21.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents an empirical study evaluating the effects of two commonly used data transformation methodologies and the L-SIAC filter when preparing high-order FEM simulation outputs (typically polynomial degree up to 6, with element-boundary discontinuities) for input to topological analysis tools designed for linear continuous data. The central claim is that these transformations produce visibly diverse behaviors in the resulting topological outputs, and the authors provide guidance on pipeline construction using currently available tools.

Significance. If the reported differences are reproducible and not artifacts of the specific test cases, the work is significant for the growing community using topological analysis on HO-FEM data. It supplies concrete evidence that preprocessing choices are not neutral and must be documented, which is a practical contribution even if the study does not claim the three methods exhaust the design space.

major comments (2)
  1. [Results] Results section: the claim of 'diverse behaviors' rests on visual comparison of topological features; without quantitative metrics (e.g., persistence diagram distances, Betti number differences, or statistical tests across multiple datasets), it is difficult to judge whether the observed differences are systematic or case-specific.
  2. [Methods] Methods: the description of the two 'commonly used' transformation methodologies lacks sufficient implementation detail (exact interpolation orders, handling of quadrature points, and boundary treatment) to permit independent reproduction or extension of the comparison.
minor comments (2)
  1. [Introduction] Abstract and introduction: the statement that 'there has been little work evaluating the possible transformation choices' would benefit from one or two additional citations to related preprocessing studies in visualization or CFD.
  2. [Figures] Figure captions: several figures showing topological outputs would be clearer if they explicitly labeled the transformation method, polynomial degree, and mesh resolution used for each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's feedback on our manuscript. The comments highlight areas where we can enhance the presentation of our empirical study. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Results] Results section: the claim of 'diverse behaviors' rests on visual comparison of topological features; without quantitative metrics (e.g., persistence diagram distances, Betti number differences, or statistical tests across multiple datasets), it is difficult to judge whether the observed differences are systematic or case-specific.

    Authors: We agree that quantitative metrics would strengthen the evaluation. In the revised manuscript we will add Betti number differences computed on the presented test cases to provide a quantitative complement to the visual comparisons. revision: yes

  2. Referee: [Methods] Methods: the description of the two 'commonly used' transformation methodologies lacks sufficient implementation detail (exact interpolation orders, handling of quadrature points, and boundary treatment) to permit independent reproduction or extension of the comparison.

    Authors: We thank the referee for noting this gap. The revised version will expand the Methods section with the requested implementation details on interpolation orders, quadrature point handling, and boundary treatment for both transformation methodologies. revision: yes

Circularity Check

0 steps flagged

No significant circularity: purely empirical comparison

full rationale

The paper conducts an empirical study comparing the effects of two common data transformation methodologies and the L-SIAC filter on topological analysis outputs for high-order FEM scalar fields. No mathematical derivations, parameter fitting, predictions from first principles, or uniqueness theorems are present. The central claim rests on direct observation of differing topological features (e.g., critical points, persistence diagrams) across the tested transformations on provided datasets. Self-citations, if any, are incidental and not load-bearing for the reported behaviors. The work is self-contained as an observational benchmark against external tools and data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an empirical comparison study; the abstract identifies no free parameters, mathematical axioms, or new postulated entities.

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