An extended finite element method with smooth nodal stress

TL;DR

This paper develops an enriched double-interpolation finite element method (DFEM) that produces smooth nodal stresses and higher-order basis functions without increasing degrees of freedom, improving accuracy and efficiency in numerical simulations.

Contribution

The paper introduces an enrichment formulation for DFEM that enhances stress smoothness and accuracy without additional degrees of freedom, extending its applicability to crack propagation problems.

Findings

DFEM achieves super-convergence and better accuracy than standard FEM.

Enriched DFEM provides more accurate stress and displacement evaluations.

Numerical examples confirm improved efficiency and precision in crack propagation simulations.

Abstract

The enrichment formulation of double-interpolation finite element method (DFEM) is developed in this paper. DFEM is first proposed by Zheng \emph{et al} (2011) and it requires two stages of interpolation to construct the trial function. The first stage of interpolation is the same as the standard finite element interpolation. Then the interpolation is reproduced by an additional procedure using the nodal values and nodal gradients which are derived from the first stage as interpolants. The re-constructed trial functions are now able to produce continuous nodal gradients, smooth nodal stress without post-processing and higher order basis without increasing the total degrees of freedom. Several benchmark numerical examples are performed to investigate accuracy and efficiency of DFEM and enriched DFEM. When compared with standard FEM, super-convergence rate and better accuracy are obtained by DFEM. For the numerical simulation of crack propagation, better accuracy is obtained in the evaluation of displacement norm, energy norm and the stress intensity factor.