r3.06.11 Hexahedral element with one integration point, stabilized by the « Assumed Strain » method

Summary:

The standard sub-integrated 8-node hexahedral element with 1 integration point introduces parasitic modes associated with zero energy (hourglass modes) and can lead to a singularity in the overall stiffness matrix for certain boundary conditions. The deficiency in the rank of the stiffness matrix, due to underintegration, must therefore be filled by adding to the elementary stiffness a so-called stabilization matrix. This is the purpose of the ASM (Assumed Strain Method) method developed here.

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.

Table of Contents

• 1. Introduction
• 2. Formulation of element HEXA8 at an integration point
  • 2.1. Displacement field and discretized gradient operator
  • 2.2. Variational formulation of the problem
  • 2.3. Hourglass modes associated with zero energy
• 3. Assumed Strain Method stabilization (ASM)
  • 3.1. Hallquist and Flanagan-Belytschko formulas
  • 3.2. Projection on a deformation field \(\stackrel{ˉ}{B}\)
  • 3.3. Choice of finite elements
• 4. Integrating the element into Code_Aster
  • 4.1. Description and use
  • 4.2. Implantation
  • 4.3. Validation
• 5. Conclusion
• 6. Bibliography
• 7. Description of document versions: