1. Introduction

In finite element calculations, the use of sub-integration methods makes it possible to significantly reduce calculation times, which explains their success. The other objective of these methods is to eliminate the various obstacles encountered in the numerical implementation of finite elements.

However, this under-integration has not only advantages: it unfortunately introduces parasitic modes associated with zero energy, which lead to hourglass modes, which will deform the mesh in an unrealistic way and end up causing the solution to explode. This is due to a deficiency in the rank of the stiffness matrix due to underintegration. This is remedied by adding a so-called stabilization matrix to the elementary stiffness. The core of the new stiffness obtained by this means must be reduced only to the modes corresponding to rigid body movements.

In recent years, some authors have developed various elements based on the ASM technique (Assumed Strain Method). The main characteristic of this method is that the discretized gradient operator \(B\) does not necessarily derive from the displacement field and from the classical relationships relating deformation to displacement. Indeed, this method ASM consists in projecting the discretized gradient operator onto an appropriate subspace in order to avoid the different types of blocking. This technique has been widely used recently and has led to several stabilized finite elements such as quadrangles with 4 knots or hexahedra with 8 knots [1], [2], [3].

It is the 8-node hexahedron element sub-integrated at 1 integration point and stabilized by the ASM method, due to Belytschko and Bindeman [2], that we describe in this document.