A Locking Free Mixed FEM Based on a Pure Pseudostress Based Formulation for the Elasticity Eigenproblem

Jul 16, 2026

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A novel locking-free mixed formulation addresses the elasticity eigenvalue problem in two and three dimensions using only the pseudostress tensor as the primary unknown. The method avoids enforcing symmetry in both weak and strong senses. The displacement field is reconstructed through postprocessing of the computed pseudostress. A mixed finite element method discretizes the H(div) space with tensor versions of standard element families, supported by convergence proofs, a priori error estimates, and non-compact operator theory.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages27

Reading Time1 h 8 min

Introduction

Motivation for mixed elasticity eigenvalue formulations

Locking-free behavior and nearly incompressible regime

Pure pseudostress formulation and discrete analysis

arXiv :2607 .06890v2 [math .NA] 9 Jul 2026 A LOCKING FREE MIXED FEM BASED ON A PURE PSEUDOSTRESS BASED FORMULATION FOR THE ELASTICITY EIGENPROBLEM ∗ ARBAZ KHAN†, FELIPE LEPE‡, AND JESUS VELLOJIN§ 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 H(div) . We prove convergence anda priori error estimates under the theory of non-compact operators. Additionally, we perform an aposteriori 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. Key words. Eigenvalue problems, finite element method, error estimates, A posteriori analysis AMS subject classifications. 35P15, 65N15, 65N25, 65N30, 74B05 1. Introduction. The accurate description of the deformation of elastic structures is crucial for the design of different devices, vehicles, buildings, etc. since their stability depends on the material properties, environments, interaction with other structures or fluids, just for mention some of the most relevant. These variables (among others of course) demand the use of reliable and robust numerical schemes for the analysis and prediction of the behavior of structures under certain conditions. On this context, we focus our attention on the linear elasticity equations, which are the most common and simple set of partial differential equations that describe the displacement of an elastic structure. On this context, we focus our attention on the description of the displacement for the eigenvalue problem associated to the elasticity equations, since this problem is related to vibration of structures which are important on the applications that we mention at the beginning of this article. The primary goal of the elasticity equations is to describe the displacement of a structure. However, other quantities of relevance may be important to also consider, such as the stress, rotations, bending moments, etc. This motivates the design of mixed formulation where additional unknowns on the PDE systems allows to describe in a more complete manner the response of a structure under certain conditions. This demands to design and analyze convenient numerical schemes capable to approximate all the unknowns involved on the mixed formulations, which are commonly called mixed methods. On this subject, the literature is abundant and particularly for ∗ Submitted to the editors DATE. Funding: Arbaz Khan was partially supported by ANRF ARG MATRICS grant ANRF/ARGM/2025/001949/MTR. Felipe Lepe was partially supported by DICREA through Proyecto Regular RE2514703, Universidad del B´ıo-B´ıo. Jesus Vellojin was partially supported by the National Agency for Research and Development, ANID-Chile through FONDECYT Postdoctorado project 3230302 . †Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India. [arbaz@ma.iitr.ac.in](arbaz@ma.iitr.ac.in) ‡GIMNAP-Departamento de Matem´atica, Universidad del B´ıo - B´ıo, Casilla 5-C, Concepci´on, [Chile. flepe@ubiobio.cl](Chile. flepe@ubiobio.cl). § Departamento de Ciencias, Universidad T´ecnica Federico Santa Mar´ıa, Av. Federico Sta. Mar´ıa 6090, Vi˜na del Mar, [Chile. jesus.vellojinm@usm.cl](Chile. jesus.vellojinm@usm.cl). 2 ARBAZ KHAN, FELIPE LEPE AND JESUS VELLOJIN the elasticity eigenvalue system we mention [4, 13 , 15 , 19 , 18 , 20 , 21] . These mixed formulations have the capability of avoiding the numerical locki

FAQs

What is the main idea of the proposed mixed formulation for elasticity eigenvalue problems?

It formulates the elasticity eigenvalue problem exclusively in terms of the pseudostress tensor, enabling a locking-free approach without requiring symmetry enforcement.

How is the displacement recovered in this formulation?

The displacement is obtained by postprocessing the computed pseudostress from the mixed finite element solution.

What does the paper claim about convergence in the nearly incompressible regime?

As the Poisson ratio approaches 1/2, the elasticity spectrum is proved to converge to the perfectly incompressible case, which matches the Stokes spectrum, using non-compact operator theory.

Jul 16, 2026

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