Low-Order Locking-Free Multiscale Finite Element Method for Isotropic Elasticity

Core Concepts

Proposing low-order finite elements for linear elasticity without Poisson locking.

Abstract

The article introduces a multiscale hybrid-mixed method for boundary value problems with heterogeneous coefficients. It presents low-order finite elements free from Poisson locking, relying on face degrees of freedom and local Neumann problems. The MHM method is well-posed, optimally convergent, and locking-free under regularity conditions. Numerical tests validate theoretical results.

Introduction to MHM methods for boundary value problems on coarse partitions.
Application of MHM to linear elasticity models using polynomial interpolations.
Stability analysis of MHM methods based on low-order global-local pairs.
Challenges with standard low-order finite element methods in nearly incompressible elasticity.
Strategies to overcome Poisson locking issues in the displacement formulation.
Stability and convergence analysis of the proposed finite elements for isotropic elasticity.

The work was partially funded by various organizations, including CNPq/Brazil and the U.S. Department of Energy.

Stats

"Two-dimensional numerical tests assess theoretical results."
"Nearly incompressible materials referred to as quasi-incompressible."
"Nearly incompressible materials have high Poisson's ratio (ν ≈ 1/2)."
"Low-order finite elements are computationally cheaper for low-regularity problems."
"Stability based on low-order global-local compromises."

Quotes

"The multiscale hybrid-mixed method approximates solutions for boundary value problems with heterogeneous coefficients."
"Low order finite elements are appealing for computationally cheaper options."
"Nearly incompressible materials exhibit poor convergence rates with standard methods."

Key Insights Distilled From

A low-order locking-free multiscale finite element method for isotropic elasticity

by Antô... at arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.16890.pdf

Deeper Inquiries

How does the choice of local level solver impact the overall stability of the MHM method

The choice of the local level solver in the Multiscale Hybrid-Mixed (MHM) method has a significant impact on its overall stability. The local level solver plays a crucial role in ensuring that the MHM method is well-posed and converges optimally under certain conditions. By selecting an appropriate local level solver, such as one that satisfies certain continuity and boundedness properties, we can guarantee the injectivity of the operator on NH, which is essential for the stability of the method. Additionally, using a Fortin operator to map functions from H1(P) to Vh ensures that Th remains injective on NH, further enhancing stability.

What are the implications of using stabilized schemes to overcome inf-sup limitations

Using stabilized schemes to overcome inf-sup limitations in numerical methods is a common approach to ensure stability and convergence. When dealing with problems where standard formulations may lead to poor convergence rates or instabilities (such as Poisson locking in nearly-incompressible elasticity problems), introducing stabilization techniques can help alleviate these issues. Stabilized schemes modify the original formulation by adding additional terms or constraints that improve stability without significantly affecting accuracy. These techniques are particularly useful when dealing with complex or challenging problems where traditional methods struggle to provide reliable solutions.

How can the concept of Poisson locking be applied to other areas outside linear elasticity

The concept of Poisson locking, commonly observed in linear elasticity problems with nearly-incompressible materials, can be applied beyond this specific domain. Poisson locking refers to poor convergence rates seen in numerical methods when approximating solutions for such materials using low-order finite elements. This phenomenon arises due to inadequate discretization strategies for handling near-incompressibility effects effectively.
In other areas outside linear elasticity, similar challenges related to locking phenomena may arise when dealing with different types of material behaviors or physical phenomena characterized by high contrast ratios or low regularity conditions. By recognizing and understanding these issues, researchers and practitioners can adapt stabilization techniques like those used in overcoming Poisson locking in elasticity problems to address similar challenges across various fields of computational science and engineering.