Stress-Hybrid Virtual Element Method for Linear Elasticity on Triangular Meshes

Core Concepts

The author presents a Stress-Hybrid Virtual Element Method for linear plane elasticity on six-noded triangular meshes, demonstrating optimal convergence and stability in the L2 norm of displacement and hydrostatic stress.

Abstract

The paper introduces a novel Stress-Hybrid Virtual Element Method (SH-VEM) for linear elasticity on triangular meshes. It adopts a unique stress-hybrid approach that ensures stability and convergence in benchmark problems. The method showcases superior accuracy compared to traditional Finite Element Methods, especially in nearly-incompressible scenarios. By incorporating penalty terms, the Penalty Stress-Hybrid VEM (PSH-VEM) further enhances performance by eliminating shear locking issues. Numerical results validate the effectiveness of both SH-VEM and PSH-VEM in various applications, showcasing their potential as advanced computational tools in engineering simulations.

Stats

The first non-rigid body eigenvalue for 15β SH-VEM is 0.63.
The first non-rigid body eigenvalue for B-bar VEM is 0.21.
The first non-rigid body eigenvalue for CT FEM is 0.81.
The penalty parameter α ranges from ℓ^2_0/10 to ℓ^2_0/100.
For the standard VEM, all five largest eigenvalues tend to infinity.
For composite triangle FEM, three largest eigenvalues diverge.
For B-bar VEM, SH-VEM, and PSH-VEM, only one large eigenvalue exists.

Quotes

"The SH-VEM converges optimally in the L2 norm of displacement and hydrostatic stress."
"PSH-VEM demonstrates faster convergence than expected optimal rate with suitable penalty parameter selection."

Key Insights Distilled From

Stress-hybrid virtual element method on six-noded triangular meshes for compressible and nearly-incompressible linear elasticity

by Alvin Chen,J... at arxiv.org 02-29-2024

https://arxiv.org/pdf/2401.06280.pdf

Deeper Inquiries

How does the introduction of penalty terms impact the overall performance of virtual element methods

The introduction of penalty terms in virtual element methods can have a significant impact on their overall performance. By incorporating penalty terms, the method can effectively address issues such as volumetric locking and shear locking that may arise in certain types of elements or under specific conditions. The penalty term helps to stabilize the formulation, ensuring that the numerical solution remains accurate and reliable even in challenging scenarios. Additionally, the use of penalty terms allows for greater flexibility in handling complex geometries and material properties, enhancing the versatility of virtual element methods.

What are the implications of using different stress basis functions on the stability and accuracy of numerical simulations

The choice of stress basis functions plays a crucial role in determining the stability and accuracy of numerical simulations using virtual element methods. Different stress basis functions can lead to varying levels of stiffness or flexibility in the elements, affecting how well they perform under different loading conditions. For example, using stress basis functions derived from Airy stress functions may provide more accurate stress distributions but could require additional higher-order terms for stability. On the other hand, hybrid formulations with a mix of linear and quadratic polynomials may offer better balance between accuracy and computational efficiency.

How can the concepts presented in this study be applied to real-world engineering problems beyond linear elasticity

The concepts presented in this study on Stress-Hybrid Virtual Element Methods (SH-VEM) and Penalty Stress-Hybrid Virtual Element Methods (PSH-VEM) have wide-ranging applications beyond linear elasticity problems. These advanced numerical techniques can be applied to real-world engineering problems across various disciplines such as structural analysis, geomechanics, fluid dynamics, electromagnetics, and more.
In structural engineering applications, SH-VEM and PSH-VEM can be utilized for analyzing complex structures subjected to different loading conditions while ensuring accurate stress predictions without suffering from locking phenomena like shear or volumetric locking.
In geotechnical engineering, these methods can help simulate soil-structure interaction problems where non-linear behavior needs to be captured accurately while maintaining computational efficiency.
Moreover, in fluid dynamics simulations involving turbulent flows or multiphase systems with intricate geometries,
the enhanced stability provided by these methods can ensure robust solutions without compromising accuracy.
These advanced virtual element techniques open up new possibilities for tackling challenging engineering problems efficiently while maintaining high levels of precision.