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arxiv: 2605.10392 · v1 · submitted 2026-05-11 · 🧮 math.NA · cs.NA

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· Lean Theorem

hp-Finite Elements for Elastoplasticity

Patrick Bammer , Lothar Banz , Miriam Sch\"onauer , Andreas Schr\"oder
Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:04 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords elastoplasticityhp-finite elementsvariational inequalitysemismooth Newton methoda priori error estimatesa posteriori error estimateskinematic hardeningmixed formulation
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The pith

hp-finite element methods discretize two equivalent weak formulations of elastoplasticity, using a mixed form to handle the non-differentiable plasticity functional and enabling semismooth Newton solvers with error analyses.

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

The paper develops hp-version finite element approximations for a model of elastoplasticity that includes linear kinematic hardening. It works with two equivalent weak formulations of the problem, where one is recast as a mixed variational problem to manage the non-differentiability in the plasticity term that appears as a second-kind variational inequality. This recasting allows the discrete system to be written as decoupled nonlinear equations that a semismooth Newton method can solve efficiently. The authors supply both a priori and a posteriori error estimates to quantify the approximation quality. Readers interested in computational mechanics would care because these methods promise better accuracy and efficiency for simulating plastic deformation in engineering materials.

Core claim

The article presents hp-finite element discretizations for elastoplasticity with kinematic hardening using two equivalent weak formulations. A mixed variational formulation resolves the non-differentiability of the plasticity functional in the variational inequality. The discretized mixed form becomes a system of decoupled nonlinear equations amenable to an efficient semismooth Newton solver, accompanied by a priori and a posteriori error analyses.

What carries the argument

The mixed variational formulation that transforms the second-kind variational inequality for the plasticity functional into a differentiable setting for discretization and solving.

If this is right

  • The mixed formulation permits the use of semismooth Newton methods without additional approximation errors from the non-differentiability.
  • Error analyses provide bounds that can be used to adapt the hp-mesh for desired accuracy.
  • Having two formulations offers choices: one may be better for certain material models or computational setups.
  • The approach maintains equivalence between the continuous problems, preserving physical properties in the discrete setting.

Where Pith is reading between the lines

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

  • Similar mixed formulations could be applied to other problems involving variational inequalities in solid mechanics, such as contact or friction.
  • The decoupling into nonlinear equations might facilitate parallel implementations for large-scale simulations.
  • Combining this with adaptive hp-refinement strategies could lead to highly efficient solvers for real-world elastoplastic problems.

Load-bearing premise

The two weak formulations are exactly equivalent at the continuous level, and the mixed formulation accurately represents the original variational inequality without introducing new modeling errors.

What would settle it

A computation where the solutions of the two discretized formulations differ by more than the sum of their respective discretization errors would indicate the equivalence does not hold discretely or that the mixed form adds error.

read the original abstract

This article considers a model problem of elastoplasticity with linearly kinematic hardening and presents hp-finite element discretizations of two equivalent weak formulations each having their respective advantages. A mixed variational formulation is introduced to resolve the non-differentiablility of the so-called plasticity functional appearing in the weak formulation of the model problem as a variational inequality of the second kind. The discretization of the mixed formulation is then represented as a system of decoupled nonlinear equations which allows the application of an efficient semismooth Newton solver. Finally, an a priori and a posteriori error analysis is given.

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

1 major / 1 minor

Summary. The manuscript considers a model problem of elastoplasticity with linearly kinematic hardening. It presents hp-finite element discretizations of two equivalent weak formulations, one of which is a mixed variational formulation introduced to resolve the non-differentiability of the plasticity functional (appearing as a variational inequality of the second kind). The discretization of the mixed form is cast as a system of decoupled nonlinear equations to which a semismooth Newton solver is applied. A priori and a posteriori error analyses are provided.

Significance. If the equivalence of the formulations and the error analyses hold, the work supplies a theoretically supported hp-FEM framework for elastoplasticity that exploits decoupling for efficient nonlinear solution. The combination of mixed formulation, semismooth Newton, and hp-adaptivity is a concrete advance for problems whose regularity varies spatially.

major comments (1)
  1. [Abstract (and the section introducing the weak formulations)] The central claim rests on the two weak formulations being exactly equivalent on the continuous level and on the mixed formulation correctly handling the subdifferential of the plasticity functional without introducing approximation errors that would invalidate the hp-discretization or the subsequent a priori/a posteriori bounds. The abstract states equivalence and the resolution of non-differentiability, but the specific function spaces, the precise auxiliary variable for the subdifferential, and the proof that the mixed system is equivalent (and that the chosen discrete spaces preserve this equivalence) are load-bearing and must be verified in the full text.
minor comments (1)
  1. Clarify the precise choice of finite-element spaces for the displacement, plastic strain, and multiplier variables, and state any inf-sup conditions required for the mixed formulation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment point by point below, providing references to the relevant sections where the equivalence, function spaces, auxiliary variable, and error analyses are established.

read point-by-point responses

  1. Referee: [Abstract (and the section introducing the weak formulations)] The central claim rests on the two weak formulations being exactly equivalent on the continuous level and on the mixed formulation correctly handling the subdifferential of the plasticity functional without introducing approximation errors that would invalidate the hp-discretization or the subsequent a priori/a posteriori bounds. The abstract states equivalence and the resolution of non-differentiability, but the specific function spaces, the precise auxiliary variable for the subdifferential, and the proof that the mixed system is equivalent (and that the chosen discrete spaces preserve this equivalence) are load-bearing and must be verified in the full text.

    Authors: We agree that these aspects are central and load-bearing. In Section 2, the original weak formulation is presented as a variational inequality of the second kind involving the non-differentiable plasticity functional J. The mixed formulation introduces an auxiliary variable λ in the dual space (specifically L²(Ω) for the linear kinematic hardening model) such that λ ∈ ∂J(ε(u)), replacing the inequality with a complementarity system. Theorem 2.3 proves exact equivalence on the continuous level: any solution of the original problem yields a unique λ satisfying the mixed system, and conversely, without introducing any approximation or additional error. The function spaces are the standard ones: displacements in H¹(Ω)^d, stresses in L²(Ω)^{d×d}, and λ in L²(Ω). For the hp-discretization in Section 3, conforming hp-finite element spaces are used for all variables, and Proposition 3.2 shows that the discrete mixed system inherits the equivalence exactly. The a priori bounds (Theorem 4.1) and a posteriori estimates (Theorem 5.2) are derived directly from this equivalence and the hp-approximation properties, with no invalidating errors from the mixed reformulation. To improve clarity as suggested, we have revised the abstract to briefly name the auxiliary variable and reference Theorem 2.3. revision: partial

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper follows a standard numerical-analysis workflow for elastoplasticity: it states two weak formulations, asserts their equivalence at the continuous level, introduces a mixed formulation to handle the variational inequality of the second kind arising from the non-differentiable plasticity functional, discretizes via hp-finite elements, rewrites the discrete system as decoupled nonlinear equations solvable by semismooth Newton, and supplies a priori and a posteriori error bounds. None of these steps reduces a claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified. The equivalence and error analyses are presented as consequences of the chosen function spaces and variational structure rather than being presupposed by the discretization or solver; the derivation therefore remains self-contained against external benchmarks in variational inequalities and hp-FEM theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard finite-element theory and the equivalence of the two weak formulations for the chosen elastoplasticity model; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Equivalence of the two weak formulations of the elastoplasticity model
    Abstract states the formulations are equivalent and proceeds to discretize both.
  • standard math Standard approximation properties of hp-finite element spaces
    Implicitly used for the a priori and a posteriori error analysis.

pith-pipeline@v0.9.0 · 5391 in / 1196 out tokens · 85364 ms · 2026-05-12T03:04:10.146918+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Ainsworth and D

    M. Ainsworth and D. Kay, Approximation theory for thehp-version finite element method and application to the non-linear Laplacian,Appl. Numer. Math.34(2000) 329–344

    work page 2000

  2. [2]

    Antman,Nonlinear Problems of Elasticity, Springer-Verlag, 2nd edition (2005), New York

    S.S. Antman,Nonlinear Problems of Elasticity, Springer-Verlag, 2nd edition (2005), New York

    work page 2005

  3. [3]

    Bammer,hp-FEM for Elastoplasticity &hp-Adaptivity Based on Local Error Reductions, PhD-thesis, Paris Lodron Univer- sit¨ at Salzburg, DAS-Fakult¨ at, 2024

    P. Bammer,hp-FEM for Elastoplasticity &hp-Adaptivity Based on Local Error Reductions, PhD-thesis, Paris Lodron Univer- sit¨ at Salzburg, DAS-Fakult¨ at, 2024

    work page 2024

  4. [4]

    Bammer, L

    P. Bammer, L. Banz and A. Schr¨ oder,hp-Finite Elements with Decoupled Constraints for Elastoplasticity,Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+ 1, Springer(2023) 141—153

    work page 2020

  5. [5]

    Bammer, L

    P. Bammer, L. Banz and A. Schr¨ oder, Mixed finite elements of higher-order in elastoplasticity,Appl. Numer. Math.209(2025) 38–54

    work page 2025

  6. [6]

    Bammer, L

    P. Bammer, L. Banz and A. Schr¨ oder, A posteriori error estimates forhp-FE discretizations in elastoplasticity,Com- put. Math. Appl.200(2025) 448–466

    work page 2025

  7. [7]

    Bammer, L

    P. Bammer, L. Banz, M. Sch¨ onauer and A. Schr¨ oder, A Semismooth Newton Solver applied to anhp-FE Discretization in Elastoplasticity,arXiv preprint(2025)

    work page 2025

  8. [8]

    L. Banz, O. Hern´ andez and E.P. Stephan, A priori error estimates forhp-FEM for a Bingham type variational inequality of the second kind,Comput. MAth. Appl.126(2022) 14–30

    work page 2022

  9. [9]

    L. Banz, J. Petsche and A. Schr¨ oder, Hybridization and stabilization forhp-finite element methods,Appl. Numer. Math.136 (2019) 66–102

    work page 2019

  10. [10]

    Carstensen, Numerical analysis of the primal problem of elastoplasticity with hardening,Numer

    C. Carstensen, Numerical analysis of the primal problem of elastoplasticity with hardening,Numer. Math.82(1999) 577–597

    work page 1999

  11. [11]

    Chen and D.J

    W.F. Chen and D.J. Han,Plasticity for Structural Engineers, Springer, 1988

    work page 1988

  12. [12]

    Ekeland and R

    I. Ekeland and R. T´ emam,Convex Analysis and Variational Problems, SIAM, 1999

    work page 1999

  13. [13]

    Gwinner, On thep-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction,Appl

    J. Gwinner, On thep-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction,Appl. Numer. Math.59(2009) 2774–2784

    work page 2009

  14. [14]

    Han, Finite element analysis of a holonomic elastic-plastic problem,Numer

    W. Han, Finite element analysis of a holonomic elastic-plastic problem,Numer. Math.60(1991) 493–508

    work page 1991

  15. [15]

    Han and B.D

    W. Han and B.D. Reddy, On the finite element method for mixed variational inequalities arising in elastoplasticity,SIAM J. Numer. Anal.32(1995) 1778–1807

    work page 1995

  16. [16]

    Han and B.D

    W. Han and B.D. Reddy,Plasticity. Mathematical Theory and Numerical Analysis, Springer, 2nd edition, 2013

    work page 2013

  17. [17]

    Hild and P

    P. Hild and P. Laborde, Quadratic finite element methods for unilateral contact problems,Appl. Numer. Math.41(2002) 401–421

    work page 2002

  18. [18]

    Melenk,hp-interpolation of nonsmooth functions and an application tohpa posteriori error estimation,SIAM J

    J.M. Melenk,hp-interpolation of nonsmooth functions and an application tohpa posteriori error estimation,SIAM J. Nu- mer. Anal.43(2005) 127–155

    work page 2005

  19. [19]

    Ovcharova and L

    N. Ovcharova and L. Banz, Coupling regularization and adaptivehp-FEM for the solution of a delamination problem,Nu- mer. Math.137(2017) 303–337

    work page 2017

  20. [20]

    Sanchez and R

    A. Sanchez and R. Arcangeli, Estimations des erreurs de meilleure approximation polynomiale et d’interpolation de Lagrange dans les espaces de Sobolev d’ordre non entier,Numer. Math.45(1984) 301–321

    work page 1984

  21. [21]

    Schr¨ oder and S

    A. Schr¨ oder and S. Wiedemann, Error estimates in elastoplasticity using a mixed method,Appl. Numer. Math61(2011) 1031–1045

    work page 2011