Mesh distortion immunity of finite elements and the best-fit paradigm

Abstract

It has been known for some time that distorted finite elements produce relatively (and, sometimes, dramatically) poor results. This has been related to the completeness condition. In this paper, we investigate this issue and propose that the abstract mathematical viewpoint represented by the completeness condition is actually a statement of the physical need for a finite element computation to recover accurate stresses in the metric space. This follows from the projection theorem describing finite element analysis which shows that the stresses computed by the displacement finite element procedure are a best approximation of the true stresses at an element as well as global level. The simplest possible element is used to elucidate the principles.