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arxiv: 2604.10796 · v1 · submitted 2026-04-12 · 🧮 math.NA · cs.NA

A DPG method for the circular arch problem

Norbert Heuer , Antti H. Niemi This is my paper

Pith reviewed 2026-05-10 15:04 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords DPG methodcircular archultra-weak formulationelasticityoptimal convergencefinite elementsbendingnumerical analysis
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The pith

The DPG method for circular arches achieves optimal convergence rates for all quantities of interest.

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

The paper presents a discontinuous Petrov-Galerkin method based on an ultra-weak formulation for modeling an elastic circular arch that includes membrane, transverse shear, and bending effects. The central line is divided into elements where discontinuous stress and displacement fields are approximated, along with interface variables at the nodes. Theoretical analysis establishes optimal convergence rates, but also identifies potential error amplification linked to the arch curvature and boundary conditions. Numerical tests on various support setups confirm these predictions and show that scaling the test space norm enhances accuracy.

Core claim

We develop an ultra-weak variational formulation for the elastic circular arch problem and a corresponding DPG approximation using optimal test functions. The formulation employs discontinuous interpolations for stresses and displacements on the element mesh with interface variables at nodes. The analysis predicts optimal convergence rates for all quantities of interest while revealing potential error amplification influenced by the curvature and the imposed boundary conditions.

What carries the argument

Ultra-weak variational formulation with DPG discretization using optimal test functions, discontinuous element interpolations, and nodal interface variables.

If this is right

  • Optimal convergence is achieved for all quantities of interest in the arch model.
  • Error amplification can occur depending on the arch curvature and boundary conditions.
  • Accuracy improves when using a suitably scaled test space norm.
  • The method works for different support configurations without additional stabilization.

Where Pith is reading between the lines

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

  • The technique may apply to other thin curved structures prone to locking phenomena in standard finite elements.
  • Automatic scaling of the test norm based on local geometry could further enhance robustness.
  • Comparative studies with other mixed finite element methods for arches would clarify relative strengths.
  • Extensions to dynamic or nonlinear arch problems could build on this stable formulation.

Load-bearing premise

The ultra-weak formulation and DPG procedure assume that the chosen discontinuous interpolations and interface variables remain stable under the specific curvature and boundary conditions of the arch model without additional stabilization beyond the scaled test norm.

What would settle it

A numerical test case where the observed convergence rates fall below the predicted optimal orders, or where error amplification does not correlate with the curvature and boundary conditions as expected, would contradict the theoretical analysis.

Figures

Figures reproduced from arXiv: 2604.10796 by Antti H. Niemi, Norbert Heuer.

Figure 1
Figure 1. Figure 1: Stability constants as functions of the curvature parameter [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Approximated displacement quantities of the cantilever circular arch problem with 8 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Approximated stress quantities of the cantilever circular arch problem with 8 elements. [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence of the full DPG solution measured by the built-in a posteriori error [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Convergence of the quantities of interested of the fully clamped arch at [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Convergence of the quantities of interested of the fully clamped arch at [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original abstract

We consider an elastic model for a circular arch that incorporates membrane, transverse shear, and bending effects. The central line of the arch is partitioned into elements, and an ultra-weak variational formulation is developed alongside a discontinuous Petrov-Galerkin (DPG) approximation procedure based on so-called optimal test functions. The formulation uses discontinuous stress and displacement interpolations on the element mesh, with corresponding interface variables defined at the nodes. Theoretical analysis predicts optimal convergence rates for all quantities of interest, while also revealing potential error amplification influenced by the curvature of the arch and the imposed boundary conditions. The method is tested on examples with different support configurations. The numerical experiments confirm the theoretical predictions and further demonstrate that the accuracy of the DPG method can be improved by employing a suitably scaled test space norm.

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

0 major / 3 minor

Summary. The paper introduces an ultra-weak variational formulation for the elastic circular arch problem, accounting for membrane, transverse shear, and bending effects. It proposes a DPG method using discontinuous interpolations for stresses and displacements on element interiors, with interface variables at nodes, and optimal test functions. The analysis predicts optimal convergence rates for all quantities of interest, highlighting potential error amplification due to arch curvature and boundary conditions. Numerical experiments on various support configurations confirm these rates and show that a suitably scaled test space norm enhances accuracy.

Significance. This work extends the DPG framework to problems involving curved geometries in structural mechanics. By providing both theoretical convergence results and numerical validation, it demonstrates the method's ability to handle the specific challenges of arch models without ad-hoc stabilization. The identification of curvature-induced error amplification is a useful insight for practitioners. The availability of numerical confirmation strengthens the practical applicability of the approach.

minor comments (3)
  1. The convergence plots in the numerical section would benefit from overlaying the predicted optimal rates (e.g., O(h^{k+1})) to facilitate direct visual verification against the theoretical predictions.
  2. The definition and choice of the scaled test space norm (mentioned in the abstract and likely detailed in the formulation section) should include an explicit statement of how the scaling parameter depends on the curvature radius to ensure reproducibility.
  3. A short comparison in the introduction with standard mixed FEM or other DPG applications to shells/arches would help readers situate the novelty of the ultra-weak formulation and interface variables.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in extending DPG methods to curved structural problems, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs an ultra-weak variational formulation for the circular arch model and applies the standard DPG method with optimal test functions. Theoretical convergence rates are derived from established DPG stability and approximation theory applied to the arch equations (membrane, shear, bending), without reducing any claimed prediction to a fitted parameter or self-referential definition. Numerical tests confirm the rates and show the effect of a scaled test norm, but this is external validation rather than a load-bearing circular step. No self-citation chain, ansatz smuggling, or renaming of known results is indicated as central to the derivation. The argument remains self-contained against external DPG benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard variational principles for elasticity and DPG theory; no free parameters or invented entities are mentioned in the abstract, though the scaled test norm may involve a tunable factor.

axioms (1)
  • domain assumption Standard assumptions of linear elasticity for the arch model hold.
    The elastic model with membrane, transverse shear, and bending effects is taken as given without derivation in the abstract.

pith-pipeline@v0.9.0 · 5425 in / 1179 out tokens · 22677 ms · 2026-05-10T15:04:05.862347+00:00 · methodology

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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    ,A DPG method for shallow shells, Numer. Math., 152 (2022), pp. 67–99. [15]J. Gopalakrishnan, I. Muga, and N. Oliv ares,Dispersive and Dissipative Errors in the DPG Method with Scaled Norms for Helmholtz Equation, SIAM J. Sci. Comput., 36 (2014), pp. A20–A39. [16]F. Kikuchi and K. Taniza w a,Accuracy and locking-free property of the beam element approxima...

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