On the equivalence between the cell-based smoothed finite element method and the virtual element method

TL;DR

This paper explores the relationship between the cell-based smoothed finite element method (SFEM) and the virtual element method (VEM), extending SFEM to complex geometries and proposing a new stabilization approach, with applications in elasticity and fracture mechanics.

Contribution

It demonstrates the equivalence between SFEM and VEM, extends SFEM to arbitrary polygons and polyhedrons, and introduces a new stabilization technique based on VEM.

Findings

SFEM and VEM are fundamentally similar methods.

The proposed stabilization improves SFEM accuracy for complex geometries.

The combined SFEM and scaled boundary method effectively handle singularities in fracture mechanics.

Abstract

We revisit the cell-based smoothed finite element method (SFEM) for quadrilateral elements and extend it to arbitrary polygons and polyhedrons in 2D and 3D, respectively. We highlight the similarity between the SFEM and the virtual element method (VEM). Based on the VEM, we propose a new stabilization approach to the SFEM when applied to arbitrary polygons and polyhedrons. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. Later, the SFEM is combined with the scaled boundary finite element method to problems involving singularity within the framework of the linear elastic fracture mechanics in 2D.