A locking free mixed FEM based on a pure pseudostress based formulation for the elasticity eigenproblem
This work addresses the need for efficient and accurate numerical methods for elasticity eigenvalue problems, particularly in nearly incompressible materials, offering a locking-free approach.
The paper presents a locking-free mixed finite element method for the elasticity eigenvalue problem using a pure pseudostress formulation, avoiding symmetry enforcement. Numerical tests validate convergence and error estimates.
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.