On the approximation in the smoothed finite element method (SFEM)

TL;DR

This paper introduces a systematic approximation scheme using non-mapped shape functions for the smoothed finite element method, addressing previous issues and enhancing its robustness especially with distorted elements.

Contribution

It proposes a new approximation scheme based on Wachspress interpolants in physical coordinates, improving SFEM's accuracy and resolving earlier deficiencies.

Findings

Results closely match original SFEM outcomes

Effective with heavily distorted elements

Provides a rigorous basis for SFEM development

Abstract

This letter aims at resolving the issues raised in the recent short communication [1] and answered by [2] by proposing a systematic approximation scheme based on non-mapped shape functions, which both allows to fully exploit the unique advantages of the smoothed finite element method (SFEM) [3, 4, 5, 6, 7, 8, 9] and resolve the existence, linearity and positivity deficiencies pointed out in [1]. We show that Wachspress interpolants [10] computed in the physical coordinate system are very well suited to the SFEM, especially when elements are heavily distorted (obtuse interior angles). The proposed approximation leads to results which are almost identical to those of the SFEM initially proposed in [3]. These results that the proposed approximation scheme forms a strong and rigorous basis for construction of smoothed finite element methods.