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arxiv: 2606.09530 · v1 · pith:3GIAAQ6Nnew · submitted 2026-06-08 · 💻 cs.CE

Reduced integration with scaled boundary parametrization for virtual elements at finite strains

Njomza Pacolli , Bjorn Sauren , Jannick Kehls , Sven Klinkel , Stefanie Reese , Hagen Holthusen This is my paper

Pith reviewed 2026-06-27 14:34 UTC · model grok-4.3

classification 💻 cs.CE
keywords virtual element methodreduced integrationscaled boundary parametrizationfinite strainsstabilizationhyperelastic anisotropyelasto-plasticitypatch test
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The pith

Scaled boundary parametrization with Taylor expansion enables one-point analytical integration to stabilize virtual elements at finite strains.

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

The paper introduces an alternative stabilization for the virtual element method at large deformations by combining reduced integration with scaled boundary parametrization. A Taylor series expansion of constitutive quantities around the sectional center permits analytical integration of the weak form using only one point per section. Numerical tests under compression, hyperelastic anisotropy, and elasto-plasticity, including a nonlinear patch test, show the method captures structural response accurately and performs better when physical elements match their parent shapes. The approach is compared against biquadratic serendipity and hourglass-stabilized finite elements and successfully handles inelastic behavior in a notched specimen.

Core claim

The formulation applies a scaled boundary parametrization together with a Taylor series expansion of the constitutive quantities with respect to the sectional center; this reduces integration of the weak form to a single analytical point per section while preserving stability and accuracy for virtual elements undergoing finite strains, hyperelastic anisotropy, and elasto-plasticity.

What carries the argument

Scaled boundary parametrization combined with Taylor series expansion of constitutive quantities around the sectional center for analytical one-point integration of the weak form.

If this is right

  • The method yields accurate results for compression under large deformations.
  • It handles hyperelastic anisotropy and elasto-plastic material behavior without loss of stability.
  • Performance improves when the physical element geometry closely matches the pre-assigned parent element.
  • Inelastic response is captured correctly in asymmetrically notched specimens.

Where Pith is reading between the lines

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

  • The single-point scheme may lower computational cost relative to standard quadrature in repeated nonlinear solves.
  • The approach could be tested on three-dimensional or higher-order virtual elements where integration expense grows rapidly.
  • Accuracy on highly distorted meshes may still require additional checks beyond the parent-element resemblance condition.

Load-bearing premise

The Taylor series expansion of the constitutive quantities with respect to the sectional center remains sufficiently accurate to allow analytical integration of the weak form with only one integration point per section.

What would settle it

A nonlinear patch test or large-deformation compression example in which the single-point analytical integration produces visible deviation from the expected uniform stress or displacement field.

Figures

Figures reproduced from arXiv: 2606.09530 by Bjorn Sauren, Hagen Holthusen, Jannick Kehls, Njomza Pacolli, Stefanie Reese, Sven Klinkel.

Figure 1
Figure 1. Figure 1: Brief overview of the presented concept of the discretization of the unknown dis [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Virtual element with its discretization along the boundary. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Considered parent elements for the stabilization term [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Isoparametric mapping and triangular sector bounded by a line element and scaling [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Application of concept of reduced integration for the stabilization term [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Non-linear patch test. Geometry and boundary value problem with the identified [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Non-linear patch test. Distribution of the first component of the second Piola [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Square block subjected to a horizontal uniform body force. Geometry and boundary [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Square block subjected to a horizontal uniform body force. Convergence study for [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Square block subjected to a horizontal uniform body force. Convergence study [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Square block subjected to a horizontal uniform body force. Contour plots of the mag [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Punch problem. Geometry and boundary value problem and the two different con [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Punch problem. Convergence study for the displacement [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Punch problem. Convergence study for the displacement [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Punch problem. Contour plots of the displacement [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Anisotropic plate with a circular hole. Geometry and boundary value problem and [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Anisotropic plate with a circular hole. Convergence study, where the sum of the [PITH_FULL_IMAGE:figures/full_fig_p028_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Anisotropic plate with a circular hole. Force-displacement curves in (a) for the [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Anisotropic plate with a circular hole. Error plot of the absolute displacement in [PITH_FULL_IMAGE:figures/full_fig_p029_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Asymmetrically notched specimen. Geometry and boundary value problem and [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Asymmetrically notched specimen. Convergence study, where the sum of the re [PITH_FULL_IMAGE:figures/full_fig_p031_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Asymmetrically notched specimen. Force-displacement curves in (a) for the finest [PITH_FULL_IMAGE:figures/full_fig_p032_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Asymmetrically notched specimen. Error plot of the absolute displacement in [PITH_FULL_IMAGE:figures/full_fig_p032_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Asymmetrically notched specimen. Contours of the accumulated plastic strain [PITH_FULL_IMAGE:figures/full_fig_p033_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: 2D block under cyclic loading. Loading curves. [PITH_FULL_IMAGE:figures/full_fig_p036_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: 2D block under cyclic loading. Force-displacement curves in (a) for a chosen [PITH_FULL_IMAGE:figures/full_fig_p037_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: 2D block under cyclic loading 1. Contours of the stabilization parameter [PITH_FULL_IMAGE:figures/full_fig_p037_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Square block subjected to a horizontal uniform body force. Convergence study for [PITH_FULL_IMAGE:figures/full_fig_p038_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Square block subjected to a horizontal uniform body force. Convergence study for [PITH_FULL_IMAGE:figures/full_fig_p039_29.png] view at source ↗
read the original abstract

This contribution presents an alternative stabilization technique for the virtual element method (VEM) based on reduced integration combined with a scaled boundary parametrization. To this end, a Taylor series expansion of the constitutive quantities with respect to the sectional center is carried out, enabling analytical integration of the weak form and reducing the need for integration points to only one per section. The accuracy of the proposed formulation is shown by several numerical examples, including a non-linear patch test. Different loading, e.g. compression under large deformations, and material conditions, such as hyperelastic anisotropy and elasto-plasticity, are considered. The biquadratic serendipity finite element formulation (Q2) and the low-order finite element formulation with hourglass stabilization (Q1STc+) are used for comparison. While the patch test was not fulfilled using higher order shape functions, the formulation led to good results and was able to capture the structure's response accurately. Furthermore, the formulation performed better when the physical element resembled the pre-assigned parent elements. The example of the asymmetrically notched specimen under elasto-plastic material behavior showed that the proposed formulation is able to capture inelasticities.

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 / 1 minor

Summary. The paper proposes an alternative stabilization for the virtual element method at finite strains based on reduced integration with scaled boundary parametrization. A Taylor series expansion of constitutive quantities about the sectional center enables analytical integration of the weak form, reducing quadrature to one point per section. Accuracy is demonstrated via numerical examples including a nonlinear patch test, large-deformation compression, hyperelastic anisotropy, and elasto-plasticity, with comparisons to Q2 and Q1STc+ finite elements; the method performs better when physical elements resemble parent elements.

Significance. If the truncation error remains controlled, the approach could supply an efficient, low-quadrature stabilization for VEM in nonlinear regimes, supported by the reported success on patch tests and inelastic examples. The numerical evidence of accurate structural response under finite strains and varied materials is a concrete strength.

major comments (2)
  1. [Formulation] Formulation (Taylor expansion step): the central claim that the expansion permits accurate single-point integration rests on the assumption that truncation of constitutive quantities about the sectional center remains sufficiently accurate at finite strains; no a priori error bound, consistency analysis, or dependence on expansion order is supplied, leaving the uncontrolled consistency error unquantified for nonlinear maps such as anisotropic hyperelasticity or elasto-plasticity.
  2. [Numerical examples] Numerical results (patch test paragraph): the abstract states the patch test 'was not fulfilled using higher order shape functions' yet 'the formulation led to good results'; this tension must be resolved by defining what constitutes fulfillment for the nonlinear case and by reporting the observed residual or convergence rate.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'the formulation performed better when the physical element resembled the pre-assigned parent elements' is presented without quantitative support or discussion of its implications for general meshes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the positive overall assessment of the work. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [Formulation] Formulation (Taylor expansion step): the central claim that the expansion permits accurate single-point integration rests on the assumption that truncation of constitutive quantities about the sectional center remains sufficiently accurate at finite strains; no a priori error bound, consistency analysis, or dependence on expansion order is supplied, leaving the uncontrolled consistency error unquantified for nonlinear maps such as anisotropic hyperelasticity or elasto-plasticity.

    Authors: We agree that the manuscript does not supply an a priori error bound or a formal consistency analysis for the first-order Taylor truncation. The approach relies on the local character of the expansion within each scaled-boundary section together with numerical verification across the reported examples. In the revised version we will add a short paragraph in the formulation section that (i) recalls the local consistency of the expansion as the section diameter tends to zero and (ii) states the order of the truncation error for the first-order expansion employed. A full a priori bound for arbitrary nonlinear constitutive laws lies beyond the scope of the present contribution but can be pursued in follow-up work. revision: yes

  2. Referee: [Numerical examples] Numerical results (patch test paragraph): the abstract states the patch test 'was not fulfilled using higher order shape functions' yet 'the formulation led to good results'; this tension must be resolved by defining what constitutes fulfillment for the nonlinear case and by reporting the observed residual or convergence rate.

    Authors: The referee correctly identifies an imprecise formulation in the abstract. In the nonlinear patch test, 'fulfillment' is understood as recovery of the exact linear displacement field up to a small residual that vanishes under mesh refinement. With higher-order shape functions the stabilization introduces a small but non-zero residual at finite resolution. We will (i) revise the abstract to remove the ambiguous phrasing, (ii) add a precise definition of the nonlinear patch-test criterion, and (iii) report the observed L2 residual norm together with the convergence rate under successive refinement in the corresponding numerical section. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard Taylor expansion on VEM weak form

full rationale

The paper derives reduced integration via Taylor expansion of constitutive quantities about the sectional center to enable one-point analytical integration per section. This is a direct application of series truncation to the weak form, independent of the target results. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided text. The non-linear patch test and numerical examples are presented as validation, not as inputs that force the outcome by construction. The central claim remains an independent stabilization technique.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.1-grok · 5754 in / 963 out tokens · 16999 ms · 2026-06-27T14:34:42.531337+00:00 · methodology

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Reference graph

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