Direct computation of stresses in linear elasticity

TL;DR

This paper introduces a finite element method for linear elasticity that directly approximates the strain tensor, demonstrating optimal convergence rates and extending previous 2D work to three dimensions.

Contribution

It presents a novel finite element approach for directly computing stresses in 3D linear elasticity, generalizing prior 2D methods with proven optimal convergence.

Findings

Demonstrates optimal convergence rate for strain tensor approximation

Extends 2D finite element methods to 3D case

Provides theoretical validation of the method's accuracy

Abstract

We present a new finite element method based on the formulation introduced by Philippe G.~Ciarlet and Patrick Ciarlet, Jr. in [{\em Math. Models Methods Appl. Sci., 15 (2005), pp. 259--571}], which approximates strain tensor directly. We also show the convergence rate of strain tensor is optimal. This work is a non-trivial generalization of its two dimensional analogue in [{\em Math. Models Methods Appl. Sci., 19 (2009), pp. 1043--1064}]