Convergence analysis of a family of 14-node brick elements

TL;DR

This paper analyzes the convergence properties of a family of 14-node brick finite elements, identifying which types achieve optimal convergence and introducing a new nonconforming element with improved accuracy.

Contribution

The paper provides a convergence analysis for Smith-Kidger 14-node elements, classifies their convergence orders, and proposes a new nonconforming element with enhanced accuracy.

Findings

Type 1, Type 2, and Type 5 elements achieve optimal convergence.

Type 6 element has lower convergence order unless modified.

Modified Type 6 element achieves optimal convergence order.

Abstract

In this paper, we will give convergence analysis for a family of 14-node elements which was proposed by I. M. Smith and D. J. Kidger in 1992. The 14 DOFs are taken as the value at the eight vertices and six face-centroids. For second-order elliptic problem, we will show that among all the Smith-Kidger 14-node elements, Type 1, Type 2 and type 5 elements can get the optimal convergence order and Type 6 get lower convergence order. Motivated by the proof, we also present a new 14-node nonconforming element. If we change the DOFs into the value at the eight vertices and the integration value of six faces, we show that Type 1, Type 2 and Type 5 keep the optimal convergence order and Type 6 element improve one order accuracy which means that it also get optimal convergence order.