A finite element method with composite shape functions

Abstract

Purpose

The purpose of this paper is to present the composite finite element method (CFEM), with Cⁿ(n≥0) continuity so it improves the accuracy of the finite element method (FEM) for solving second‐order partial differential equations (PDEs) and also, can be used for solving higher order PDEs.

Design/methodology/approach

In this method, the nodal values in the conventional FEM have been replaced by the appropriate nodal functions. Based on this idea, a procedure has been proposed for obtaining the CFEM−Cⁿ shape functions based on the CFEM−Cⁿ⁻¹ shape functions as follows: the nodal values in the CFEM−Cⁿ⁻¹ have been replaced by deliberately selected nodal functions so that the smoothness of the CFEM−Cⁿ⁻¹ shape functions increase.

Findings

The proposed method has the following properties: first, its shape functions have simple explicit forms with respect to the natural coordinates of elements and consequently, the required integrals for calculation of stiffness matrix can be evaluated numerically by low‐order Gauss quadratures; second, numerical investigations show that the CFEM with Cⁿ(n>1) continuity leads to more accurate results in comparison with the FEM; third, in multi‐dimensional problems, the curved boundaries are modeled more accurately by the proposed method in comparison with the FEM; fourth, this method can treat with the weak discontinuities such as the interface between different materials, as simple as the FEM does; and fifth, this method can successfully model Kirchhoff plate problems.

Originality/value

This method is an improvement of the moving particle FEM and reproducing kernel element method. Despite these two methods, CFEM shape functions have simple explicit forms with respect to the natural coordinates of elements.

Citation

Sadeghirad, A. and Vaziri Astaneh, A. (2011), "A finite element method with composite shape functions", Engineering Computations, Vol. 28 No. 4, pp. 389-423. https://doi.org/10.1108/02644401111131867