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A locking free mixed FEM based on a pure pseudostress based formulation for the elasticity eigenproblem

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Pith's one-line read A pure-pseudostress mixed finite-element method for elasticity eigenvalues is locking-free for any Poisson ratio, recovers displacement by post-processing, and comes with a residual estimator that is reliable and efficient independently of

desk verdict Solid pure-pseudostress mixed FEM for the elasticity eigenproblem; locking-free rates rest on one flagged regularity assumption. read the letter →

arxiv 2607.06890 v2 pith:OEZZEZKV submitted 2026-07-08 math.NA cs.NA

Arbaz Khan , Felipe Lepe , Jesus Vellojin This is my paper · ORCID
classification math.NAcs.NA MSC 35P1565N1565N2565N3074B05
keywords elasticityeigenvalueproblempseudostressformulationlocking-freemixedFEMRaviart–Thomaselementsnon-compactoperatorsa-posteriorierrorestimationnearlyincompressiblematerials
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a mixed variational formulation of the linear-elasticity eigenvalue problem whose only unknown is the pseudostress tensor. Because the formulation never enforces symmetry of the stress and never involves the displacement as a primary variable, standard Raviart–Thomas elements can be used without locking when the material becomes nearly or perfectly incompressible. The continuous solution operator is non-compact, so spectral convergence and optimal a-priori rates for eigenvalues and eigenfunctions are proved by the theory of non-compact operators. As the Lamé parameter tends to infinity the spectrum converges to that of the Stokes eigenvalue problem, recovering the incompressible limit. A residual-based a-posteriori indicator is shown to be both reliable and efficient uniformly in the compressibility parameter; adaptive meshes driven by this indicator restore optimal rates on non-convex domains in two and three dimensions. Displacement and true stress are recovered by elementary post-processing of the computed pseudostress.

What carries the argument

The self-adjoint non-compact solution operator T_λ that maps a load to the unique pseudostress solving the shifted source problem; its spectral decomposition and the discrete counterpart T_λ,h allow the application of the Descloux–Nassif–Rappaz theory to obtain locking-free a-priori estimates.

What would settle it

Compute the first few eigenvalues on a sequence of uniformly refined meshes for Poisson ratios successively closer to 1/2 (or for the formal limit λ=∞) and check whether the observed orders of convergence for both eigenvalues and eigenfunctions remain exactly those predicted by the a-priori theory; any systematic degradation would falsify the locking-free claim.

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Extended reading notes

Core claim

A pure-pseudostress formulation of the elasticity eigenproblem, discretized by tensor-valued Raviart–Thomas elements, is locking-free: eigenvalues and eigenfunctions converge at the rates predicted by the regularity of the eigenfunctions, the spectrum of the nearly-incompressible operator converges to the Stokes spectrum, and a residual estimator remains reliable and efficient independently of the Lamé parameter.

Load-bearing premise

The Sobolev regularity of the eigenfunctions and the constant that bounds it are assumed independent of the Lamé parameter, even though the paper itself notes that this independence is not completely evident from the analysis.

Editorial extensions

If this is right

  • Standard H(div)-conforming elements can be used for vibration analysis of nearly-incompressible elastic bodies without artificial stiffening.
  • The spectrum of the discrete elasticity operator converges to the Stokes spectrum as the Poisson ratio approaches 1/2, giving a practical route to incompressible eigencomputations.
  • The residual estimator can drive adaptive refinement that recovers optimal rates on domains with re-entrant corners or edges, uniformly in compressibility.
  • Displacement and true Cauchy stress are obtained by simple post-processing of the pseudostress, so no additional mixed system needs to be solved.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same pure-pseudostress idea should extend, with only minor changes, to related non-self-adjoint or damped eigenvalue problems in viscoelasticity.
  • Because the formulation never requires symmetry, it is a natural candidate for hybridization or static condensation that would further reduce the algebraic cost of three-dimensional computations.
  • The residual estimator’s independence of λ suggests it could serve as a reliable stopping criterion in iterative solvers that themselves become ill-conditioned near the incompressible limit.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes a pure-pseudostress mixed variational formulation of the linear elasticity eigenvalue problem (2.5)–(2.10) that avoids any symmetry constraint on the tensor. The associated solution operator T_λ is non-compact; spectral characterization (Theorem 2.3), convergence of the nearly-incompressible spectrum to the Stokes spectrum (Lemma 2.8, Theorem 2.9), and discrete analysis via Raviart–Thomas elements are carried out with the Descloux–Nassif–Rappaz theory. Optimal a-priori rates for eigenfunctions and double-order rates for eigenvalues (Theorems 4.4–4.5) are proved under an explicit regularity assumption independent of λ. A residual a-posteriori estimator η (and its incompressible counterpart) is shown reliable and efficient with λ-independent weights (Theorems 5.1, 5.4). Numerical experiments on convex and non-convex domains in 2D/3D confirm locking-free convergence and adaptive recovery of optimal rates.

Significance. A locking-free pure-pseudostress eigenvalue formulation that never enforces symmetry is a useful addition to the mixed-FEM literature for elasticity eigenproblems. The analysis is self-contained once standard Sobolev regularity is granted, the residual estimator is new for this setting, and the numerical tests (including adaptive refinement near re-entrant corners and the incompressible limit) are reproducible and support the claims. If Assumption 2.6 holds, the method supplies a practical, symmetry-free alternative to existing mixed schemes that remains robust as ν→1/2.

major comments (2)
  1. Assumption 2.6 (independence of the regularity exponent r and constant Ĉ of Lemma 2.4 with respect to λ) is load-bearing for the locking-free rates of Theorems 4.4–4.5 and for the claim that the estimator constants are independent of λ. The manuscript itself notes that the independence “is not completely evident” and is adopted only because numerics still show the expected orders at λ=∞. Either a reference establishing λ-uniform regularity for the pure-pseudostress source problem, or a short remark quantifying possible deterioration of the constants, is needed before the locking-free claim can be regarded as fully rigorous.
  2. Reliability (Theorem 5.1) is stated with an extra term ∥κρ−κ_h ρ_h∥_0 + ∥ρ−ρ_h∥_0 on the right-hand side; efficiency (Theorem 5.4) absorbs a generic higher-order remainder Θ. For the estimator to be fully practical one needs either a proof that these terms are of higher order (or controlled by η itself) or a clear statement that they are neglected only after the discrete eigenpair has already converged. The present wording leaves a small gap between the proved bounds and the quantity that is actually used for marking.
minor comments (4)
  1. Several typographical slips appear: “the discrete the eigenvalue problem” (heading 3.2), “let su assume” (p. 16), “con the perfectly incompressible case” (abstract), and inconsistent boldface for tensor/vector fields.
  2. The definition of the weights ρ_R and ρ_E (and their incompressible analogues) is dense; a short table or displayed list would improve readability of Section 5.1.
  3. In the numerical section the effectivity index is defined as err(κ_i)/η²; a brief justification why the square appears (consistent with the double-order eigenvalue estimate) would help the reader.
  4. References [13,15,20,22] are heavily used for background lemmas; a one-sentence pointer to the precise statements that are imported would make the paper more self-contained for non-specialists.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pure-pseudostress formulation, non-compact spectral theory, locking-free rates and residual estimator are derived self-containedly from standard mixed-FEM arguments once Assumption 2.6 is granted.

full rationale

The paper constructs the pure-pseudostress eigenvalue problem (2.5)–(2.10), introduces the non-compact solution operator T_λ, characterises its spectrum (Theorem 2.3), proves operator convergence to the incompressible limit T_∞ (Lemma 2.8, Theorem 2.9), establishes discrete spectral convergence via Descloux–Nassif–Rappaz properties P1–P2 (Lemmas 3.1, 4.1–4.3) and obtains the a-priori rates of Theorems 4.4–4.5 by the usual gap and algebraic-identity arguments. The residual estimator is built from the strong form of the discrete residual and proved reliable/efficient by standard bubble-function techniques (Theorems 5.1, 5.4). All steps are either elementary (Lax–Milgram, Céa, integration by parts) or cite classical external references ([5], [7], [8], [1], [24]). Self-citations to earlier mixed formulations by overlapping authors supply background lemmas that are independently published and are not used as the sole justification of any central rate. Assumption 2.6 (λ-independence of the regularity exponent) is explicitly flagged as an assumption motivated by numerics; it is not derived circularly from the claimed rates. No parameter is fitted and then re-labelled a prediction, no uniqueness theorem is imported solely from the authors’ prior work to force the present choice, and no known empirical pattern is merely renamed. The derivation is therefore free of the six circularity patterns.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The paper rests on standard Sobolev and mixed-method theory plus one explicit regularity assumption whose λ-independence is not proved. No free parameters are fitted; the only invented objects are the pure-pseudostress formulation itself and the residual indicator, both of which are fully defined inside the paper.

assumptions (3)
  • ad hoc to paper Assumption 2.6: the regularity exponent r and the constant Ĉ of Lemma 2.4 are independent of the Lamé parameter λ.
    Invoked to obtain locking-free rates; the paper itself remarks that independence “is not completely evident” and is supported only by numerics.
  • standard math Standard H(div) approximation properties of the Raviart–Thomas interpolant and the L2 projector (Section 3.1).
    Classical results used for Céa-type estimates and for the discrete compactness argument.
  • domain assumption Existence of a positive regularity exponent s∈(0,1] for the dual mixed elasticity source problem (estimate (2.17)).
    Taken from the literature on mixed elasticity; needed for compactness of T_λ restricted to P(X0).
invented entities (2)
  • Pure-pseudostress variational formulation (2.9)–(2.10) without symmetry constraint
    purpose: Eliminate both displacement and the symmetry constraint from the eigenvalue problem while remaining locking-free.
    Defined entirely inside the paper; no external existence claim is made beyond the analysis given.
  • Residual error indicator η (and its incompressible counterpart η_∞) of Section 5
    purpose: Drive adaptive mesh refinement that remains reliable and efficient as ν→1/2.
    Constructed from element and jump residuals of the discrete pseudostress; reliability/efficiency proved inside the paper.

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Pith. "Pith review of A locking free mixed FEM based on a pure pseudostress based formulation for the elasticity eigenproblem." pith.science (2026). https://pith.science/paper/OEZZEZKV

@misc{pith2026260706890,
  author       = {Pith},
  title        = {Pith review of: A locking free mixed FEM based on a pure pseudostress based formulation for the elasticity eigenproblem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OEZZEZKV}},
  note         = {Machine review of arXiv:2607.06890}
}
abstract

We analyze a novel locking-free mixed formulation for the elasticity eigenvalue problem in both two and three dimensions, expressed exclusively in terms of the pseudostress tensor. An important feature of this formulation is that it does not require the enforcement of symmetry, either in a weak or strong sense. The displacement of the structure is recovered via a postprocess of the computed pseudostress. We introduce a mixed finite element method based in the tensorial version of the standard families of finite elements to discretize the space $\boldsymbol{\mathcal{H}}(\bdiv)$. We prove convergence and a priori error estimates under the theory of non-compact operators. Additionally, we perform an a posteriori error analysis for the problem, proving reliability and efficiency of the proposed indicator. We validate our theoretical results with numerical tests on different geometrical and physical configurations.

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